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#### Question 1:

stion Prove the following identities (1-16)
sec4 x sec2 x = tan4 x + tan2 x

#### Question 2:

Prove the following identities (1-16)

#### Question 3:

Prove the following identities (1-16)

#### Question 4:

Prove the following identities (1-16)

#### Question 5:

Prove the following identities (1-16)

#### Question 6:

Prove the following identities (1-16)

#### Question 7:

Prove the following identities (1-16)

#### Question 8:

Prove the following identities (1-16)

#### Question 9:

Prove the following identities (1-16)

#### Question 10:

Prove the following identities (1-16)

#### Question 11:

Prove the following identities (1-16)

= RHS
Hence proved.

#### Question 12:

Prove the following identities (1-16)

= RHS

Hence proved.

#### Question 13:

Prove the following identities (1-17)

Hence proved.

#### Question 14:

Prove the following identities (1-16)

#### Question 15:

Prove the following identities (1-16)

#### Question 16:

Prove the following identities (1-16)

#### Question 17:

If , then prove that is also equal to a.

Disclaimer: There is some error in the given question.
The question should have been
Question: If , then prove that is also equal to a.
So, the solution is done accordingly.

Solution:

Hence proved.

#### Question 18:

If , then the values of tan x, sec x and cosec x

$\mathrm{tan}x=\frac{\mathrm{sin}x}{\mathrm{cos}x}=\frac{\frac{{a}^{2}-{b}^{2}}{{a}^{2}+{b}^{2}}}{\frac{2ab}{{a}^{2}+{b}^{2}}}=\frac{{a}^{2}-{b}^{2}}{2ab}\phantom{\rule{0ex}{0ex}}\mathrm{sec}x=\frac{1}{\mathrm{cos}x}=\frac{{a}^{2}+{b}^{2}}{2ab}\phantom{\rule{0ex}{0ex}}\mathrm{cosec}x=\frac{1}{\mathrm{sin}x}=\frac{{a}^{2}+{b}^{2}}{{a}^{2}-{b}^{2}}$

#### Question 19:

If , then find the values of $\sqrt{\frac{a+b}{a-b}}+\sqrt{\frac{a-b}{a+b}}$.

If show that .

Hence proved.

#### Question 21:

If , then prove that .

#### Question 22:

If prove that ${\left({m}^{2}+{n}^{2}\right)}^{2}=mn$.

#### Question 23:

If , then prove that , where ${m}^{2}\le 2$

#### Question 24:

If , then shown that $ab+a-b+1=0$.

Hence proved.

Prove the:

#### Question 26:

If , prove that

(i) $\frac{{T}_{3}-{T}_{5}}{{T}_{1}}=\frac{{T}_{5}-{T}_{7}}{{T}_{3}}$

(ii)

(iii)

(i) LHS:

RHS:
$\frac{{T}_{5}-{T}_{7}}{{T}_{3}}\phantom{\rule{0ex}{0ex}}=\frac{\left({\mathrm{sin}}^{5}x+{\mathrm{cos}}^{5}x\right)-\left({\mathrm{sin}}^{7}x+{\mathrm{cos}}^{7}x\right)}{{\mathrm{sin}}^{3}x+{\mathrm{cos}}^{3}x}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{{\mathrm{sin}}^{5}x-\mathrm{si}{n}^{7}x+{\mathrm{cos}}^{5}x-{\mathrm{cos}}^{7}x}{{\mathrm{sin}}^{3}x+{\mathrm{cos}}^{3}x}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{{\mathrm{sin}}^{5}x\left(1-{\mathrm{sin}}^{2}x\right)+{\mathrm{cos}}^{5}x\left(1-{\mathrm{cos}}^{2}x\right)}{{\mathrm{sin}}^{3}x+{\mathrm{cos}}^{3}x}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{{\mathrm{sin}}^{5}x{\mathrm{cos}}^{2}x+{\mathrm{cos}}^{5}x{\mathrm{sin}}^{2}x}{{\mathrm{sin}}^{3}x+{\mathrm{cos}}^{3}x}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}={\mathrm{sin}}^{2}x.{\mathrm{cos}}^{2}x$

LHS = RHS

Hence proved.

(ii) LHS:

Hence proved.

(iii) LHS:

$6{T}_{10}-15{T}_{8}+10{T}_{6}-1\phantom{\rule{0ex}{0ex}}6\left({\mathrm{sin}}^{10}x+{\mathrm{cos}}^{10}x\right)-15\left({\mathrm{sin}}^{8}x+{\mathrm{cos}}^{8}x\right)+10\left({\mathrm{sin}}^{6}x+{\mathrm{cos}}^{6}x\right)-1\phantom{\rule{0ex}{0ex}}$

#### Question 1:

Find the values of the other five trigonometric functions in each of the following:

#### Question 2:

If sin $x=\frac{12}{13}$ and x lies in the second quadrant, find the value of sec x + tan x.

#### Question 3:

If sin  find the value of 8 tan .

We have:

#### Question 4:

If sin x + cos x = 0 and x lies in the fourth quadrant, find sin x and cos x.

#### Question 5:

If  find the values of other five trigonometric functions and hence evaluate .

#### Question 1:

Find the values of the following trigonometric ratios:
(i) $\mathrm{sin}\frac{5\mathrm{\pi }}{3}$

(ii) sin 17π

(iii) $\mathrm{tan}\frac{11\mathrm{\pi }}{6}$

(iv) $\mathrm{cos}\left(-\frac{25\mathrm{\pi }}{4}\right)$

(v)

(vi) $\mathrm{sin}\frac{17\pi }{6}$

(vii) $\mathrm{cos}\frac{19\pi }{6}$

(viii) $\mathrm{sin}\left(-\frac{11\pi }{6}\right)$

(ix) $\mathrm{cosec}\left(-\frac{20\pi }{3}\right)$

(x) $\mathrm{tan}\left(-\frac{13\pi }{4}\right)$

(xi) $\mathrm{cos}\frac{19\pi }{4}$

(xii) $\mathrm{sin}\frac{41\pi }{4}$

(xiii) $\mathrm{cos}\frac{39\pi }{4}$

(xiv) $\mathrm{sin}\frac{151\pi }{6}$

#### Question 2:

Prove that:
(i) tan 225° cot 405° + tan 765° cot 675° = 0
(ii) $\mathrm{sin}\frac{8\mathrm{\pi }}{3}\mathrm{cos}\frac{23\mathrm{\pi }}{6}+\mathrm{cos}\frac{13\mathrm{\pi }}{3}\mathrm{sin}\frac{35\mathrm{\pi }}{6}=\frac{1}{2}$
(iii) cos 24° + cos 55° + cos 125° + cos 204° + cos 300° = $\frac{1}{2}$
(iv) tan (−225°) cot (−405°) −tan (−765°) cot (675°) = 0
(v) cos 570° sin 510° + sin (−330°) cos (−390°) = 0
(vi) $\mathrm{tan}\frac{11\mathrm{\pi }}{3}-2\mathrm{sin}\frac{4\mathrm{\pi }}{6}-\frac{3}{4}{\mathrm{cosec}}^{2}\frac{\mathrm{\pi }}{4}+4{\mathrm{cos}}^{2}\frac{17\mathrm{\pi }}{6}=\frac{3-4\sqrt{3}}{2}$
(vii) $3\mathrm{sin}\frac{\mathrm{\pi }}{6}\mathrm{sec}\frac{\mathrm{\pi }}{3}-4\mathrm{sin}\frac{5\mathrm{\pi }}{6}\mathrm{cot}\frac{\mathrm{\pi }}{4}=1$

#### Question 3:

Prove that

(i)

(ii)  $\frac{\mathrm{cosec}\left(90°+x\right)+\mathrm{cot}\left(450°+x\right)}{\mathrm{cosec}\left(90°-x\right)+\mathrm{tan}\left(180°-x\right)}+\frac{\mathrm{tan}\left(180°+x\right)+\mathrm{sec}\left(180°-x\right)}{\mathrm{tan}\left(360°+x\right)-\mathrm{sec}\left(-x\right)}=2$

(iii)

(iv)

(v)

#### Question 4:

Prove that: ${\mathrm{sin}}^{2}\frac{\mathrm{\pi }}{18}+{\mathrm{sin}}^{2}\frac{\mathrm{\pi }}{9}+{\mathrm{sin}}^{2}\frac{7\mathrm{\pi }}{18}+{\mathrm{sin}}^{2}\frac{4\mathrm{\pi }}{9}=2$

#### Question 5:

Prove that:
$\mathrm{sec}\left(\frac{3\mathrm{\pi }}{2}-x\right)\mathrm{sec}\left(x-\frac{5\mathrm{\pi }}{2}\right)+\mathrm{tan}\left(\frac{5\mathrm{\pi }}{2}+x\right)\mathrm{tan}\left(x-\frac{3\mathrm{\pi }}{2}\right)=-1.$

#### Question 6:

In a ∆ABC, prove that:
(i) cos (A + B) + cos C = 0
(ii) $\mathrm{cos}\left(\frac{A+B}{2}\right)=\mathrm{sin}\frac{C}{2}$
(iii) $\mathrm{tan}\frac{A+B}{2}=\mathrm{cot}\frac{C}{2}$

#### Question 7:

In a ∆A, B, C, D be the angles of a cyclic quadrilateral, taken in order, prove that
cos(180° − A) + cos (180° + B) + cos (180° + C) − sin (90° + D) = 0

#### Question 8:

Find x from the following equations:

$90°=\frac{\pi }{2}$

#### Question 9:

Prove that:
(i)

(ii) $\mathrm{sin}\frac{13\pi }{3}\mathrm{sin}\frac{8\pi }{3}+\mathrm{cos}\frac{2\pi }{3}\mathrm{sin}\frac{5\pi }{6}=\frac{1}{2}$

(iii)

(iv) $\mathrm{sin}\frac{10\pi }{3}\mathrm{cos}\frac{13\pi }{6}+\mathrm{cos}\frac{8\pi }{3}\mathrm{sin}\frac{5\pi }{6}=-1$

(v) $\mathrm{tan}\frac{5\mathrm{\pi }}{4}\mathrm{cot}\frac{9\mathrm{\pi }}{4}+\mathrm{tan}\frac{17\mathrm{\pi }}{4}\mathrm{cot}\frac{15\mathrm{\pi }}{4}=0$

#### Question 1:

If tan x = $x-\frac{1}{4x}$, then sec x − tan x is equal to
(a) $-2x,\frac{1}{2x}$
(b) $-\frac{1}{2x},2x$
(c) 2x
(d) $2x,\frac{1}{2x}$

(a) $-2x,\frac{1}{2x}$

#### Question 2:

If sec $x=\mathrm{x}+\frac{1}{4\mathrm{x}}$, then sec x + tan x =
(a) $x,\frac{1}{x}$
(b) $2x,\frac{1}{2x}$
(c) $-2x,\frac{1}{2x}$
(d) $-\frac{1}{x},x$

(b) $2x,\frac{1}{2x}$

#### Question 3:

If  is equal to
(a) sec x − tan x
(b) sec x + tan x
(c) tan x − sec x
(d) none of these

(c) tan x − sec x

#### Question 4:

If π < x <2π, then  is equal to
(a) cosec x + cot x
(b) cosec x − cot x
(c) −cosec x + cot x
(d) −cosec x − cot x

(d) −cosec x − cot x

#### Question 5:

If $0, and if , then y is equal to
(a) $\mathrm{cot}\frac{x}{2}$

(b) $\mathrm{tan}\frac{x}{2}$

(c) $\mathrm{cot}\frac{x}{2}+\mathrm{tan}\frac{x}{2}$

(d) $\mathrm{cot}\frac{x}{2}-\mathrm{tan}\frac{x}{2}$

(b) $\mathrm{tan}\frac{x}{2}$

If  is equal to
(a) 2 sec x
(b) −2 sec x
(c) sec x
(d) −sec x

(b)  −2 sec x

#### Question 7:

If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then x2 + y2 + z2 is independent of
(a) θ, ϕ
(b) r, θ
(c) r, ϕ
(d) r.

(a) θ, ϕ

We have:
x = r sin θ cos ϕ  ,  y = r sin θ sin ϕ and z = r cos θ,
x2 + y2 + z2

#### Question 8:

If tan x + sec x = $\sqrt{3}$, 0 < x < π, then x is equal to

(a) $\frac{5\mathrm{\pi }}{6}$

(b) $\frac{2\mathrm{\pi }}{3}$

(c) $\frac{\mathrm{\pi }}{6}$

(d) $\frac{\mathrm{\pi }}{3}$

(c) $\frac{\mathrm{\pi }}{6}$

#### Question 9:

If tan $x=-\frac{1}{\sqrt{5}}$ and θ lies in the IV quadrant, then the value of cos x is
(a) $\frac{\sqrt{5}}{\sqrt{6}}$

(b) $\frac{2}{\sqrt{6}}$

(c) $\frac{1}{2}$

(d) $\frac{1}{\sqrt{6}}$

(a) $\frac{\sqrt{5}}{\sqrt{6}}$

If  is equal to
(a)  1 − cot α
(b) 1 + cot α
(c) −1 + cot α
(d) −1 −cot α

(d) −1 −cot α

#### Question 11:

sin6 A + cos6 A + 3 sin2 A cos2 A =
(a) 0
(b) 1
(c) 2
(d) 3

(b) 1

#### Question 12:

If , then cos x is equal to
(a) $\frac{5}{3}$

(b) $\frac{3}{5}$

(c) $-\frac{3}{5}$

(d) $-\frac{5}{3}$

(b) $\frac{3}{5}$

#### Question 13:

If , then tan x =
(a) $\frac{21}{22}$

(b) $\frac{15}{16}$

(c) $\frac{44}{117}$

(d) $\frac{117}{44}$

(c) $\frac{44}{117}$

#### Question 14:

${\mathrm{sec}}^{2}x=\frac{4xy}{\left(x+y{\right)}^{2}}$ is true if and only if
(a) x + y ≠ 0
(b) x = y, x ≠ 0
(c) x = y
(d) x ≠0, y ≠ 0

(b) x = y, x ≠ 0

#### Question 15:

If x is an acute angle and , then the value of  is
(a) 3/4
(b) 1/2
(c) 2
(d) 5/4

(a) 3/4

#### Question 16:

The value of sin25° + sin210° + sin215° + ... + sin285° + sin290° is
(a) 7
(b) 8
(c) 9.5
(d) 10

(c) 9.5

#### Question 17:

sin2 π/18 + sin2 π/9 + sin2 7π/18 + sin2 4π/9 =
(a) 1
(b) 4
(c) 2
(d) 0

(c) 2

#### Question 18:

If tan A + cot A = 4, then tan4 A + cot4 A is equal to
(a) 110
(b) 191
(c) 80
(d) 194

(d) 194

#### Question 19:

If x sin 45° cos2 60° = , then x =
(a) 2
(b) 4
(c) 8
(d) 16

(c) 8

#### Question 20:

If A lies in second quadrant 3tanA + 4 = 0, then the value of 2cotA − 5cosA + sinA is equal to

(a) $-\frac{53}{10}$                           (b) $\frac{23}{10}$                           (c) $\frac{37}{10}$                           (d) $\frac{7}{10}$

It is given that $\frac{\mathrm{\pi }}{2}.

$3\mathrm{tan}A+4=0\phantom{\rule{0ex}{0ex}}⇒\mathrm{tan}A=-\frac{4}{3}\phantom{\rule{0ex}{0ex}}⇒\mathrm{cot}A=-\frac{3}{4}$

Now,

Also,

So,

$2\mathrm{cot}A-5\mathrm{cos}A+\mathrm{sin}A\phantom{\rule{0ex}{0ex}}=2×\left(-\frac{3}{4}\right)-5×\left(-\frac{3}{5}\right)+\frac{4}{5}\phantom{\rule{0ex}{0ex}}=-\frac{3}{2}+3+\frac{4}{5}\phantom{\rule{0ex}{0ex}}=\frac{-15+30+8}{10}\phantom{\rule{0ex}{0ex}}=\frac{23}{10}$

Hence, the correct answer is option B.

#### Question 21:

If , then tan x =
(a) $\frac{21}{22}$

(b) $\frac{15}{16}$

(c) $\frac{44}{117}$

(d) $\frac{117}{43}$

(c) $\frac{44}{117}$

#### Question 22:

If tan θ + sec θ =ex, then cos θ equals

(a) $\frac{{e}^{x}+{e}^{-x}}{2}$
(b) $\frac{2}{{e}^{x}+{e}^{-x}}$
(c) $\frac{{e}^{x}-{e}^{-x}}{2}$
(d) $\frac{{e}^{x}-{e}^{-x}}{{e}^{x}+{e}^{-x}}$

(b) $\frac{2}{{e}^{x}+{e}^{-x}}$

#### Question 23:

If sec x + tan x = k, cos x =

(a) $\frac{{k}^{2}+1}{2k}$

(b) $\frac{2k}{{k}^{2}+1}$

(c) $\frac{k}{{k}^{2}+1}$

(d) $\frac{k}{{k}^{2}-1}$

(b) $\frac{2k}{{k}^{2}+1}$

#### Question 24:

If $f\left(x\right)={\mathrm{cos}}^{2}x+{\mathrm{sec}}^{2}x$, then

(a) f(x) < 1                             (b) f(x) = 1                             (c) 1 < f(x) < 2                              (d) f(x) ≥ 2

Hence, the correct option is answer D.

#### Question 25:

Which of the following is incorrect?

(a)                 (b) cos x = 1                (c)                 (d) tan x = 20

(a) is correct as

(b) cos x = 1 is correct as

(c) is not correct as

(d) tan x = 20 is correct as tan x can take any real value.

Hence, the correct answer is option C.

#### Question 26:

The value of is

(a) $\frac{1}{\sqrt{2}}$                         (b) 0                          (c) 1                          (d) $-1$

Hence, the correct answer is option B.

#### Question 27:

The value of is

(a) 0                                  (b) 1                                  (c) $\frac{1}{2}$                                  (d) not defined

We know that, $\mathrm{tan}\left(90°-\theta \right)=\mathrm{cot}\theta$

So,

Hence, the correct answer is option B.

#### Question 28:

Which of the following is correct?

(a) $\mathrm{sin}1°>\mathrm{sin}1$                    (b) $\mathrm{sin}1°<\mathrm{sin}1$                    (c) $\mathrm{sin}1°=\mathrm{sin}1$                    (d) $\mathrm{sin}1°=\frac{\mathrm{\pi }}{180}\mathrm{sin}1$

We know that, 1 radian is approximately 57º.

Also, the value of sinx is always increasing for $0\le x\le 90°$ ( or sinx is an increasing function for $0\le x\le 90°$).

Now,

Hence, the correct answer is option B.

#### Question 29:

If then sin θ is equal to

(a) $-\frac{4}{5}$ but not $\frac{4}{5}$

(b) $-\frac{4}{5}$ or $\frac{4}{5}$

(c) $\frac{4}{5}$ but not $-\frac{4}{5}$

(d) none of these

Given
Since tan is negative in 2nd or 4th quadrant
⇒ θ lies in 2nd or 4th quadrant Since AC2 = AB2 + BC2 = (–4)2 + (3)2
AC2 = 25
AC = ± 5
i.e. AC = 5

Hence, the correct answer is option B.

#### Question 30:

If sin θ and cos θ are the roots of the equation ax2bx + c = 0, then a, b and c satisfy the relation
(a) a2 + b2 + 2ac = 0
(b) a2b2 + 2ac = 0
(c) a2 + c2 + 2ab = 0
(d) a2b2 – 2ac = 0

Given sinθ and cosθ are roots of ax2 – bx + c = 0
Sum of  roots is $\frac{b}{a}$ and product of root is $\frac{c}{a}$
i.e
Since sin2θ + cos2θ = 1

Hence, the correct answer is option B.

#### Question 31:

If sin θ + cosec θ = 2, then sin2θ + cosec2θ is equal to
(a) 1
(b) 4
(c) 2
(d) none of these

Given sinθ + cosecθ = 2
⇒ (sinθ + cosecθ)2 = 4
i.e sin2θ + cosec2θ + 2 sinθ  cosecθ = 4

Hence, the correct answer is option C.

#### Question 32:

Which of the following is incorrect?
(a) $\mathrm{sin}\theta =-\frac{1}{5}$
(b) cosθ = 1
(c) $\mathrm{sec}\theta =\frac{1}{2}$
(d) tanθ = 20

i.e No solution

Hence, the correct answer is option C.

#### Question 33:

If for real values of  then
(a) θ is an acute angle
(b) θ is a right angle
(c) θ is an obtuse angle
(d) No value of θ is possible

Given for real value of x$\mathrm{cos}\theta =x+\frac{1}{x}$

which is not possible        (∵ –1 ≤ cosθ ≤ 1)
∴ No such value of θ is possible
Hence, the correct answer is option D.

#### Question 1:

The value of is _____________.

#### Question 2:

The value of is _____________.

#### Question 3:

The values of lie in the interval __________ .

#### Question 4:

If sin x + cos x = a, then sin x – cos x = __________.

Given sin x + cos x = a
i.e (sin x – cos x)=a2           (squaring both side)

Now, consider

#### Question 5:

If $-\frac{\mathrm{\pi }}{2} then $\sqrt{\frac{1-\mathrm{sin}x}{1+\mathrm{sin}x}}$ is equal to _________.

If $-\frac{\mathrm{\pi }}{2} is given,

#### Question 6:

If  then sec x – tan x = ___________.

Given

#### Question 7:

If  then tan x = ___________.

Subtracting (1) from (2)
$\mathrm{sec}x+\mathrm{tan}x-\mathrm{sec}x+\mathrm{tan}x=\frac{3}{2}-\frac{2}{3}\phantom{\rule{0ex}{0ex}}⇒2\mathrm{tan}x=\frac{9-4}{6}\phantom{\rule{0ex}{0ex}}⇒2\mathrm{tan}x=\frac{5}{6}\phantom{\rule{0ex}{0ex}}⇒\mathrm{tan}x=\frac{5}{12}$

#### Question 8:

If cosec x + cot x = α, then sin x = _________.

Given cosec x + cot x =         (1)
Since cosec2x – cot2x = 1
(cosec x – cot x) (cosec x + cot x) = 1
i.e (cosec x – cot x) a = 1

#### Question 9:

If then the value of tan x is ______________.

#### Question 10:

If  then the value of tan x is __________ .

Given  i.e x lies in III or IV quadrant #### Question 11:

If sin x + cosec x = 2, then sin2x + cosec2x = ___________ .

Given sin x + cosec x = 2
Squaring both sides, (sin x + cosec x)2 = 4

#### Question 12:

The value of tan  is ______________ .

#### Question 13:

The value of 3 (sin x – cos x)4 + 6 (sin x + cos x)2 + 4 (sin6x + cos6x) is ___________.

3 (sin x – cos x)4 + 6 (sin x + cos x)2 + 4 (sin6x + cos6x)

#### Question 14:

Given x > 0, the value of  lie in the interval ____________.

Given x > 0,

#### Question 15:

If sin x + cos x = a, then sin6x + cos6x = __________.

Given sin x + cos x = a
Squaring both sides, (sin x + cos x)2 = a2

Using identity, we have

Hence, sin6x + cos6x

#### Question 16:

The value of cos 1° cos 2° cos 3° _______cos 179° is ____________.

cos 1° cos 2° cos 3° ....... cos 179°
= cos 1° cos 2° cos 3° ......... cos 90° cos 91° ......... cos 179°
= cos 1° × cos 2° × cos 3° ..... × 0 × cos 91° .......... cos 179°    (∵ cos 90° = 0)
= 0 × finite
= 0
Hence, cos 1° cos 2° cos 3° ......... cos 179° = 0

#### Question 17:

The value of tan 1° tan 2° tan 3° _________ tan 89° is _________.

tan 1° tan 2° tan 3° ........ tan 89°
Since 89° = 90° – 1
88° = 90° – 2 ........... 46° = 90° – 44°
and tan 89° = tan (90° – 1°) = cot 1°
tan 88° = tan (90° – 2°) = cot 2°
.
.
.
tan 46° = tan (90° – 44°) = cot 44°
tan 45° = 1
∴ tan 1° tan 2° tan 3° .......... tan 45° cot 44° .....cot 3° cot 2° cot 1°

= 1
Hence, tan 1° tan 2° ....... tan 89° = 1.

#### Question 18:

If then k = ___________.

If $\frac{\mathrm{\pi }}{2}
Given

(correction)

#### Question 19:

If then k = ___________.

#### Question 20:

If then k = ___________.

If π < x < 2π

#### Question 21:

The minimum value of 9 tan2θ + 4 cot2θ is ____________.

9 tan2θ + 4 cot2θ
Since Arithmetic mass ≥ Geometric mean for 2 tans.

i.e minimum value of 9 tan2θ + 4 cot2θ  is 12.

#### Question 22:

If sec x = m and tan x = n, then $\frac{1}{m}\left\{\left(m+n\right)+\frac{1}{m+n}\right\}$ is equal to ____________.

If sec x = m and tan x = n

#### Question 23:

If cos2x + sin x + 1 = 0, and 0 < x < 2π then x = _________.

Given,

#### Question 24:

If  and x lies in the second quadrant, then cos x = ___________.

Given  and x lies in 2nd quadrant
Since sin2x + cos2x = 1
cos2x = 1 – sin2x
Since x lies in II Quadrant
⇒ cos x < 0

#### Question 25:

If  then the value of sec x + tan x is _________.

#### Question 26:

If tan x + cot x = 4, then tan4x + cot4x = ___________.

Given tan x + cot x = 4

#### Question 27:

If $\frac{3\mathrm{\pi }}{4} then  is equal to ___________.

Given $\frac{3\mathrm{\pi }}{4}

#### Question 1:

Write the maximum and minimum values of cos (cos x).

#### Question 2:

Write the maximum and minimum values of sin (sin x).

#### Question 3:

Write the maximum value of sin (cos x).

#### Question 4:

If sin x = cos2 x, then write the value of cos2 x (1 + cos2 x).

#### Question 5:

If sin x + cosec x = 2, then write the value of sinn x + cosecn x.

#### Question 6:

If sin x + sin2 x = 1, then write the value of cos12 x + 3 cos10 x + 3 cos8 x + cos6 x.

#### Question 7:

If sin x + sin2 x = 1, then write the value of cos8 x + 2 cos6 x + cos4 x.

#### Question 8:

If sin θ1 + sin θ2 + sin θ3 = 3, then write the value of cos θ1 + cos θ2 + cos θ3.

Sine function can take the maximum value of 1.
If, $\mathrm{sin}{\theta }_{1}+\mathrm{sin}{\theta }_{2}+\mathrm{sin}{\theta }_{3}=3$, then we have:

sin${\theta }_{1}$ = 1

⇒ ${\theta }_{1}$=$\frac{\mathrm{\pi }}{2}$
Similarly, ${\theta }_{2}={\theta }_{3}=\frac{\mathrm{\pi }}{2}$

$⇒\mathrm{cos}{\theta }_{1}=\mathrm{cos}{\theta }_{2}=\mathrm{cos}{\theta }_{3}=0\phantom{\rule{0ex}{0ex}}⇒\mathrm{cos}{\theta }_{1}+\mathrm{cos}{\theta }_{2}+\mathrm{cos}{\theta }_{3}=0$

#### Question 9:

Write the value of sin 10° + sin 20° + sin 30° + ... + sin 360°.

#### Question 10:

A circular wire of radius 15 cm is cut and bent so as to lie along the circumference of a loop of radius 120 cm. Write the measure of the angle subtended by it at the centre of the loop.

Circumference of the circle of radius 15 cm:

Now, 94.2 cm will be the length of arc$\left(\mathrm{l}\right)$ for the circle with radius 120 cm.
We know:

45$°$ = $\frac{\mathrm{\pi }}{4}=\frac{22}{7×4}=0.785$ radians
Therefore, the angle subtended by it at the centre of the loop is 45$°$.

#### Question 11:

Write the value of 2 (sin6 x + cos6 x) −3 (sin4 x + cos4 x) + 1.

$2\left({\mathrm{sin}}^{6}x+{\mathrm{cos}}^{6}x\right)-3\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)+1\phantom{\rule{0ex}{0ex}}=2\left({\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x\right)\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x-{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x\right)-3\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)+1\phantom{\rule{0ex}{0ex}}=2.1\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x-{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x\right)-3\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)+1\phantom{\rule{0ex}{0ex}}=2\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)-2{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x-3\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)+1\phantom{\rule{0ex}{0ex}}=-\left({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x\right)-2{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x+1\phantom{\rule{0ex}{0ex}}=-\left\{{\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x+2{\mathrm{sin}}^{2}x.c{\mathrm{os}}^{2}x\right\}+1\phantom{\rule{0ex}{0ex}}=-{\left({\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x\right)}^{2}+1\phantom{\rule{0ex}{0ex}}=-1+1\phantom{\rule{0ex}{0ex}}=0$

#### Question 12:

Write the value of cos 1° + cos 2° + cos 3° + ... + cos 180°.

#### Question 13:

If cot (α + β) = 0, then write the value of sin (α + 2β).

#### Question 14:

If tan A + cot A = 4, then write the value of tan4 A + cot4 A.

#### Question 15:

Write the least value of cos2 x + sec2 x.

We know:
cos x can take the minimum value of $-1$.

cos2 x + sec2 x

$=\frac{{\mathrm{cos}}^{4}x+1}{{\mathrm{cos}}^{2}x}\phantom{\rule{0ex}{0ex}}=\frac{{\left(-1\right)}^{4}+1}{{\left(-1\right)}^{2}}\phantom{\rule{0ex}{0ex}}=2$

#### Question 16:

If x = sin14 x + cos20  x, then write the smallest interval in which the value of x lie.

If x = 0$°$, 90$°$, 180$°$, 270$°$, 360$°$, then

The smallest interval in which the value of x lie is $\left(0,1\right]$.

#### Question 17:

If 3 sin x + 5 cos x = 5, then write the value of 5 sin x − 3 cos x.