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  4. In a Δ ABC , ( cos 2((B-C/2))/(b+c)2)+( sin 2((B-C/2))/(b-c)2)=

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Q. In a $\Delta ABC , \frac{\cos ^2\left(\frac{B-C}{2}\right)}{(b+c)^2}+\frac{\sin ^2\left(\frac{B-C}{2}\right)}{(b-c)^2}=$

TS EAMCET 2021

Solution:

$\frac{\cos ^2\left(\frac{B-C}{2}\right)}{(b+c)^2}+\frac{\sin ^2\left(\frac{B-C}{2}\right)}{(b-c)^2}$
$ \begin{Bmatrix}\text { by using sine theorem } \\ \frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{k} \text { (let) }\end{Bmatrix}$
$ (b+c)^2=k^2(\sin B+\sin C)^2$
$ =4 k^2 \sin ^2\left(\frac{B+C}{2}\right) \cos ^2\left(\frac{B-C}{2}\right) $
$ \Rightarrow 4 k^2 \sin ^2\left(\frac{180-A}{2}\right) \cos ^2\left(\frac{B-C}{2}\right) $
$ \Rightarrow \frac{(b+c)^2}{\cos ^2\left(\frac{B-C}{2}\right)}=4 k^2 \cos ^2\left(\frac{A}{2}\right) $
$ \Rightarrow \frac{1}{4 k^2 \cos ^2\left(\frac{A}{2}\right)}+\frac{1}{4 k^2 \sin ^2\left(\frac{A}{2}\right)} $
$ \Rightarrow \frac{\cos ^2\left(\frac{A}{2}\right)+\sin ^2\left(\frac{A}{2}\right)}{4 k^2 \cos ^2 \frac{A}{2} \sin ^2 \frac{A}{2}} $
$ \Rightarrow \frac{1}{\left(2 k \sin ^2 \cos \frac{A}{2}\right)^2} \Rightarrow \frac{1}{k^2 \sin ^2 A}=\frac{1}{a^2}$
$ a=k \sin ^A $
$ S O, \frac{1}{a^2}$