Question

If $\tan \frac{x}{2}=\cos ecx+\sin x$, then ${\sec }^2\frac{x}{2}$ is equal to

Answer 1

$\tan \left(\frac{x}{2}\right)=cosecx+\sin x$

$\Rightarrow {\tan }^2\left(\frac{x}{2}\right)=cose{c}^{2}x+{\sin }^{2}x+2cosecx\times \sin x$

$\Rightarrow {\tan }^2\left(\frac{x}{2}\right)=\frac{1}{{\sin }^{2}x}+{\sin }^2+x+2\times \frac{1}{\left(\sin x\right)}\left(\sin x\right)$

$\Rightarrow 1+{\tan }^2\left(\frac{x}{2}\right)=\frac{1}{{\sin }^{2}x}+{\sin }^{2}x+3$

$\Rightarrow {\sec }^2\left(\frac{x}{2}\right)=cose{c}^{2}x+{\sin }^{2}x+3$

(Since ${\tan }^{2}x+1={\sec }^{2}x)$