Question

can anyone say urgent pls write with steps . i will like your answer and mark u as a brainlist if wrong reported ​

Answer 1

Step-by-step explanation:

Let the angles of the given triangle be

α,β,γ i.e., A = α, B = β, C = γ

Now, we have,

$\sum \frac{ \tan( \frac{ \alpha }{2} ) }{(s - b)(s - c)}$

$= \frac{ \tan( \frac{ \alpha }{2} ) }{(s - b)(s - c)} + \frac{ \tan( \frac{ \beta }{2} ) }{(s - c)(s - a)} +\frac{ \tan( \frac{ \gamma }{2} ) }{(s - a)(s - b)}$

We know that,

$\tan( \frac{ \alpha }{2} ) = \sqrt{ \frac{(s - b)(s - c)}{s(s - a)} }$

$= \frac{ \sqrt{ \frac{(s - b)(s - c)}{s(s - a)} } }{(s - b)(s - c)} + \frac{ \sqrt{ \frac{(s - c)(s - a)}{s(s - b)} } }{(s - c)(s - a)} + \frac{ \sqrt{ \frac{(s - a)(s - b)}{s(s - c)} } }{(s - a)(s - b)}$

$= \frac{ \sqrt{(s - b)(s - c)} }{(s - b)(s - c) \sqrt{ s(s - a)}} + \frac{ \sqrt{(s - c)(s - a)} }{(s - c)(s - a) \sqrt{ s(s - b)}} +\frac{ \sqrt{(s - a)(s - b)} }{(s - a)(s - b) \sqrt{ s(s - c)}}$

$= \frac{1}{ \sqrt{s(s - a)(s - b)(s - c)} } + \frac{1}{ \sqrt{s(s - a)(s - b)(s - c)} } + \frac{1}{ \sqrt{s(s - a)(s - b)(s - c)} }$

$= \frac{3}{ \triangle}$