Page No 9.28:
Question 1:
Prove that:
Answer:
Page No 9.28:
Question 2:
Prove that:
Answer:
Page No 9.28:
Question 3:
Prove that:
Answer:
Page No 9.28:
Question 4:
Prove that:
Answer:
Page No 9.28:
Question 5:
Prove that:
Answer:
Page No 9.28:
Question 6:
Prove that:
Answer:
Page No 9.28:
Question 7:
Prove that:
Answer:
On dividing the numerator and denominator by cos
x, we get
Page No 9.28:
Question 8:
Prove that:
Answer:
On dividing the numerator and denominator by , we get
Page No 9.28:
Question 9:
Prove that:
Answer:
Page No 9.28:
Question 10:
Prove that:
Answer:
Page No 9.28:
Question 11:
Prove that:
Answer:
Page No 9.28:
Question 12:
Prove that:
Answer:
Using the identity
, we get
Page No 9.28:
Question 13:
Prove that:
Answer:
Using the identity
, we get
Page No 9.28:
Question 14:
Prove that:
Answer:
Using the identity
, we get
Page No 9.28:
Question 15:
Prove that:
Answer:
Using the identities
, we get
Page No 9.28:
Question 16:
Prove that:
Answer:
Page No 9.28:
Question 17:
Prove that:
Answer:
Page No 9.28:
Question 18:
Prove that:
Answer:
Now, using the identities
, we get
Page No 9.28:
Question 19:
Show that:
Answer:
Page No 9.28:
Question 20:
Show that:
Answer:
Page No 9.28:
Question 21:
Prove that:
Answer:
Page No 9.28:
Question 22:
Prove that:
Answer:
Page No 9.28:
Question 23:
Prove that:
Answer:
Page No 9.28:
Question 24:
Prove that:
Answer:
Page No 9.28:
Question 25:
Prove that
Answer:
Page No 9.29:
Question 26:
Answer:
Page No 9.29:
Question 27:
Prove that:
Answer:
Page No 9.29:
Question 28:
(i) If and x lies in the IIIrd quadrant, find the values of
.
(ii) If and x lies in IInd quadrant, find the values of sin 2x and
Answer:
(i)
Using the identity , we get
It is given that x lies in the third quadrant. This means that lies in the second quadrant.
Again,
It is given that x lies in the third quadrant. This means that lies in the second quadrant.
Since x lies in the third quadrant, sinx is negative.
(ii)
Here, x lies in the second quadrant.
We know,
sin2x = 2sinx cosx
Now,
Since x lies in the second quadrant, lies in the first quadrant.
Page No 9.29:
Question 29:
If and x lies in IInd quadrant, find the values of
Answer:
Given:
Using the identity , we get
Since x lies in the 2nd quadrant, cosx is negative.
Now,
Since x lies in the 2nd quadrant, lies in the 1st quadrant.
Again,
Page No 9.29:
Question 30:
(i) If 0 ≤ x ≤ π and x lies in the IInd quadrant such that . Find the values of
(ii) If and x is acute, find tan 2x
(iii) If and , find the value of sin 4x.
Answer:
(i)
Since x lies in the 2nd quadrant, cos x is negative.
Thus,
Now, using the identity , we get
Since x lies in the 2nd quadrant and lies in the 1st quadrant, is positive.
Again,
Now,
(ii)
Now,
Hence, the value of tan 2x is .
(iii) and .
Since x lies in the 1st quadrant, cos x is positive.
Thus,
Now,
Hence, the value of sin 4x is .
Page No 9.29:
Question 31:
If , then find the value of . [NCERT]
Answer:
Given:
Page No 9.29:
Question 32:
If and , show that cos 2A = sin 4B
Answer:
Given:
and
Using the identity , we get
Now, using the identities , we get
∴ cos 2A = sin 4B
Page No 9.29:
Question 33:
Prove that:
Answer:
On dividing and multiplying by
, we get
Page No 9.29:
Question 34:
Prove that:
Answer:
On dividing and multiplying by
, we get
Page No 9.29:
Question 35:
Prove that:
Answer:
Page No 9.29:
Question 36:
Prove that:
Answer:
On dividing and multiplying by
, we get
Page No 9.29:
Question 37:
If prove that
Answer:
Given:
Page No 9.29:
Question 38:
If , prove that
(i)
(ii)
Answer:
The given equations are .
(i)
Now, using the identity for the LHS of, we get
On dividing (1) by (2), we get
We know,
(ii)
Page No 9.29:
Question 39:
If , prove that
Answer:
Page No 9.29:
Question 40:
If , prove that
Answer:
Given:
...(1)
Page No 9.29:
Question 41:
If , prove that
Answer:
Equation can be written as
Page No 9.30:
Question 42:
If and sin , prove that
Answer:
Squaring and adding equations and , we get
Now,
Page No 9.30:
Question 43:
If , prove that
Answer:
Given:
.
Now,
Now,
Page No 9.30:
Question 44:
If has α and β as its roots, then prove that
(i) [NCERT EXEMPLAR]
(ii)
(iii) [NCERT EXEMPLAR]
Answer:
Given:
This a quadratic equation in terms of .
It is given that α and β are the roots of the given equation, so tanα and tanβ are the roots of (1).
Since tanα and tanβ are the roots of the equation . Therefore,
(i)
(ii)
(iii)
Page No 9.30:
Question 45:
If , then prove that . [NCERT EXEMPLAR]
Answer:
Given:
Squaring on both sides, we get
Also,
Squaring on both sides, we get
Subtracting (2) from (1), we get
Page No 9.36:
Question 1:
Prove that:
Answer:
Page No 9.36:
Question 2:
Prove that:
Answer:
Page No 9.36:
Question 3:
Prove that:
Answer:
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Question 4:
Page No 9.36:
Question 5:
Answer:
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Question 6:
Answer:
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Question 7:
Answer:
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Question 8:
Answer:
Page No 9.37:
Question 9:
Answer:
Page No 9.37:
Question 10:
Prove that for all values of x
Answer:
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Question 11:
Prove that for all values of x
Answer:
Page No 9.42:
Question 1:
Prove that:
Answer:
Page No 9.42:
Question 2:
Prove that:
Answer:
Page No 9.42:
Question 3:
Prove that:
Answer:
Page No 9.42:
Question 4:
Prove that:
Answer:
Page No 9.42:
Question 5:
Prove that:
Answer:
Hence proved.
Page No 9.42:
Question 6:
Prove that:
Answer:
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Question 7:
Prove that:
Answer:
Page No 9.42:
Question 8:
Prove that:
Answer:
Page No 9.42:
Question 9:
Prove that :
Answer:
Thus, LHS = RHS
Hence,
.
Page No 9.42:
Question 10:
Prove that:
Answer:
Page No 9.42:
Question 1:
is equal to
(a) 8 cos
x
(b) cos
x
(c) 8 sin
x
(d) sin
x
Answer:
(d) sin x
Page No 9.42:
Question 2:
(a)
(b)
(c)
(d) none of these.
Answer:
(b)
Page No 9.42:
Question 3:
The value of is
(a)
(b)
(c)
(d) none of these
Answer:
(d) none of these
Page No 9.42:
Question 4:
If then, is equal to
(a) 1
(b)
(c)
(d)
Answer:
(a) 1
Page No 9.42:
Question 5:
For all real values of x, is equal to
(a)
(b)
(c)
(d) none of these
Answer:
(b)
Page No 9.42:
Question 6:
The value of is
(a) 0
(b)
(c) 1
(d) none of these
Answer:
(a) 0
Page No 9.42:
Question 7:
If in a , then
(a) 6
(b) 1
(c)
(d) none of these
Answer:
(d) none of these
ABC is a triangle.
Page No 9.42:
Question 8:
If and , then
(a)
(b)
(c) 1
(d) none of these
Answer:
(b)
Page No 9.42:
Question 9:
If
(a)
(b)
(c)
(d) none of these
Answer:
(a)
Page No 9.43:
Question 10:
If , then
(a)
(b)
(c)
(d) none of these
Answer:
(b)
Page No 9.43:
Question 11:
If
(a)
(b)
(c)
(d) none of these
Answer:
(b)
Page No 9.43:
Question 12:
The value of is
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(d) 4
Page No 9.43:
Question 13:
The value of
(a) 1
(b)
(c)
(d) none of these.
Answer:
(d) none of these
Page No 9.43:
Question 14:
(a) 1
(b) 2
(c) 4
(d) none of these.
Answer:
(b) 2
Page No 9.43:
Question 15:
If , then is equal to
(a)
(b)
(c)
(d)
Answer:
(c)
Page No 9.43:
Question 16:
The value of is
(a) 2
(b) 1
(c) 0
(d) −1
Answer:
(c) 0
Page No 9.43:
Question 17:
If , then A lies in the interval
(a)
(b)
(c)
(d) none of these
Answer:
(a)
Page No 9.43:
Question 18:
The value of is equal to
(a) cos x
(b) sin x
(c) tan x
(d) none of these
Answer:
(a) cos x
Page No 9.43:
Question 19:
If
(a) 3
(b) 4
(c) 1
(d) 2
Answer:
(d) 2
Page No 9.43:
Question 20:
The value of is
(a)
(b) 0
(c)
(d)
Answer:
(a)
Page No 9.43:
Question 21:
is equal to
(a) cos
x
(b) sin
x
(c) – cos
x
(d) sin
x
Answer:
(b) sin x
Page No 9.43:
Question 22:
The value of is
(a) 0
(b) cos 3A
(c) cos 2A
(d) none of these
Answer:
(c) cos 2A
Page No 9.44:
Question 23:
The value of is
(a) cos x
(b) sec x
(c) cosec x
(d) sin x
Answer:
(c) cosec x
Page No 9.44:
Question 24:
is equal to
(a)
(b)
(c)
(d)
Answer:
(a)
Page No 9.44:
Question 25:
If α and β are acute angles satisfying , then tan α =
(a)
(b)
(c)
(d)
Answer:
(a)
Page No 9.44:
Question 26:
If , then
(a)
(b)
(c)
(d)
Answer:
(d)
Page No 9.44:
Question 27:
If then
(a)
(b) 1
(c) 1/2
(d) None of these
Answer:
(b) 1
Page No 9.44:
Question 28:
If then
(a)
(b)
(c)
(d)
Answer:
(a)
Page No 9.44:
Question 29:
The value of is
(a) cos 2x
(b) sin 2x
(c) cos 4x
(d) none of these
Answer:
(c) cos 4x
Page No 9.44:
Question 30:
The value of is
(a) cos 2A
(b) sin 2A
(c) cos A
(d) 0
Answer:
(a) cos 2A
Page No 9.44:
Question 31:
The value of is
(a) cot 3x
(b) 2cot 3x
(c) tan 3x
(d) 3 tan 3x
Answer:
(c) tan 3x
Page No 9.44:
Question 32:
The value of is
(a) 3 tan 3x
(b) tan 3x
(c) 3 cot 3x
(d) cot 3x
Answer:
(a) 3 tan 3x
Page No 9.44:
Question 33:
The value of
(a)
(b)
(c)
(d) None of these
Answer:
(c)
Page No 9.44:
Question 34:
is equal to
(a)
(b)
(c)
(d)
Answer:
(a)
Page No 9.44:
Question 35:
If is equal to
(a)
(b)
(c)
(d)
(e) None of these
Answer:
(d)
Page No 9.45:
Question 36:
If , then is equal to
(a) a
(b) b
(c)
(d)
Answer:
Given:
Now,
Hence, the correct answer is option B.
Given:
Now,
Hence, the correct answer is option B.
Page No 9.45:
Question 37:
If , then is equal to
(a) (b) (c) (d)
Answer:
It is given that and .
Now,
Hence, the correct answer is option B.
Page No 9.45:
Question 38:
The value of is
(a) (b) (c) (d)
Answer:
Hence, the correct answer is option A.
Page No 9.45:
Question 39:
The value of is
(a) 1
(b)
(c)
(d) 2
Answer:
Hence, the correct answer is option C.
Page No 9.45:
Question 40:
The value of tan 75° – cot75° is
(a)
(b)
(c)
(d) 1
Answer:
Page No 9.45:
Question 41:
cos 2θ cos 2Ï + sin2(θ – Ï) – sin2(θ + Ï) is equal to
(a) sin 2 (θ + Ï)
(b) cos 2 (θ + Ï)
(c) sin 2 (θ – Ï)
(d) cos 2 (θ – Ï)
Answer:
Page No 9.45:
Question 42:
If then tan (2A + B) is equal
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
Page No 9.45:
Question 43:
If sin θ + cos θ = 1, then the value of sin 2θ is equal to
(a) 1
(b)
(c) 0
(d) –1
Answer:
Page No 9.45:
Question 44:
The value of is
(a)
(b)
(c)
(d) 1
Answer:
Page No 9.45:
Question 45:
If and θ lies in third quadrant, then the value of is
(a)
(b)
(c)
(d)
Answer:
Page No 9.46:
Question 46:
The value of cos 12° + cos 84° + cos 156° + cos 132° is
(a)
(b) 1
(c)
(d)
Answer:
Page No 9.46:
Question 1:
If then k = __________.
Answer:
Page No 9.46:
Question 2:
If cos x cos 2x cos22x______cos2n – 1 then λ = ______________.
Answer:
Page No 9.46:
Question 3:
The value of is ___________.
Answer:
Page No 9.46:
Question 4:
If then tan 2x = _____________.
Answer:
Page No 9.46:
Question 5:
If then the numerical value of k is __________.
Answer:
Page No 9.46:
Question 6:
In a triangle ABC with the equation whose roots are tan A and tan B is.
Answer:
Page No 9.46:
Question 7:
The value of cos248° – sin212° is ___________.
Answer:
Page No 9.46:
Question 8:
The least value of 2 sin2θ + 3cos2θ is ___________.
Answer:
Page No 9.46:
Question 9:
If cos6x + sin6x + k sin22x = 1, then k = ___________.
Answer:
Page No 9.46:
Question 10:
The value of is ____________.
Answer:
Page No 9.46:
Question 11:
The value of is _____________.
Answer:
Page No 9.46:
Question 12:
If tan θ = t, then tan 2θ + sec 2θ = ___________.
Answer:
Page No 9.46:
Question 13:
If then asin 2θ + b cos 2θ is equal to __________.
Answer:
Page No 9.46:
Question 14:
If and cos2x = sin ky, then k = ______________.
Answer:
Page No 9.46:
Question 15:
The value of is ______________.
Answer:
Page No 9.46:
Question 16:
The value of is ______________.
Answer:
Page No 9.46:
Question 17:
The value of cos26° – cos224° is ___________.
Answer:
Page No 9.47:
Question 18:
If then ______________.
Answer:
Page No 9.47:
Question 19:
If then ______________.
Answer:
Page No 9.47:
Question 20:
The value of 108 is ____________.
Answer:
Page No 9.47:
Question 1:
If , then write the value of k.
Answer:
Page No 9.47:
Question 2:
If , then write the value of m sin x + n cos x.
Answer:
Given:
Page No 9.47:
Question 3:
If , then write the value of .
Answer:
Page No 9.47:
Question 4:
If the write the value of in the simplest form.
Answer:
We have,
Page No 9.47:
Question 5:
If , then write the value of .
Answer:
We have,
Page No 9.47:
Question 6:
If , then write the value of .
Answer:
Page No 9.47:
Question 7:
In a right angled triangle ABC, write the value of sin2A + Sin2B + Sin2C.
Answer:
Page No 9.47:
Question 8:
Write the value of .
Answer:
Page No 9.47:
Question 9:
If , then write the value of .
Answer:
Page No 9.47:
Question 10:
Write the value of
Answer:
Proceeding in the same way, we get
Page No 9.47:
Question 11:
If , then find the value of tan2A.
Answer:
Given:
Hence, the value of tan2A is tanB.
Page No 9.47:
Question 12:
If , then find the value of .
Answer:
Given:
Squaring on both sides, we get
Now,
Hence, the required value is .
Page No 9.47:
Question 13:
If , find the value of .
Answer:
Given:
Now,
Thus, the required value is .
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