cotA2+cotB2+cotC2=ss-as-bs-c+ss-bs-as-c+ss-cs-as-b=ss-as-bs-c×ss-ass-a+ss-bs-as-c×ss-bss-b+ss-cs-as-b×ss-css-c=ss-ass-as-bs-c+ss-bss-as-bs-c+ss-css-as-bs-c=ss-a△+ss-b△+ss-c△=s△3s-a-b-c=s△3s-2s=s2△Hence,cotA2+cotB2+cotC2=s2△Note:Consider △ with sides a,b,c then Semi-perimeter,s=a+b+c2 and Area of triangle,△=ss-as-bs-c

Solve the question 9

\cot \frac{A}{2} + \cot \frac{B}{2} + \cot \frac{C}{2} = \sqrt{\frac{s (s - a)}{(s - b) (s - c)}} + \sqrt{\frac{s (s - b)}{(s - a) (s - c)}} + \sqrt{\frac{s (s - c)}{(s - a) (s - b)}} = \sqrt{\frac{s (s - a)}{(s - b) (s - c)} \times \frac{s (s - a)}{s (s - a)}} + \sqrt{\frac{s (s - b)}{(s - a) (s - c)} \times \frac{s (s - b)}{s (s - b)}} + \sqrt{\frac{s (s - c)}{(s - a) (s - b)} \times \frac{s (s - c)}{s (s - c)}} = \frac{s (s - a)}{\sqrt{s (s - a) (s - b) (s - c)}} + \frac{s (s - b)}{\sqrt{s (s - a) (s - b) (s - c)}} + \frac{s (s - c)}{\sqrt{s (s - a) (s - b) (s - c)}} = \frac{s (s - a)}{△} + \frac{s (s - b)}{△} + \frac{s (s - c)}{△} = \frac{s}{△} (3 s - a - b - c) = \frac{s}{△} (3 s - 2 s) = \frac{s^{2}}{△} Hence, \cot \frac{A}{2} + \cot \frac{B}{2} + \cot \frac{C}{2} = \frac{s^{2}}{△} Note : Consider △ with sides a, b, c then Semi - perimeter, s = \frac{a + b + c}{2} and Area of triangle, △ = \sqrt{s (s - a) (s - b) (s - c)}

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