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  4. General solution of equation 1+2 operatornamecosec x=-( sec 2 (x/2)/2) is

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Q. General solution of equation $1+2 \operatorname{cosec} x=-\frac{\sec ^2 \frac{x}{2}}{2}$ is

Trigonometric Functions

Solution:

$1+2 \operatorname{cosec} x=\frac{-\sec ^2\left(\frac{x}{2}\right)}{2}$
$ \Rightarrow 1+\frac{2}{\sin x}=\frac{-1}{1+\cos x}$
$\Rightarrow(2+\sin x)(1+\cos x)=-\sin x$
$ \Rightarrow 2+2 \cos x+\sin x+\sin x \cos x=-\sin x$
$\Rightarrow 2(\sin x+\cos x)+\sin x \cos x+2=0$
Put $\sin x+\cos x=t$
$\Rightarrow 1+2 \sin x \cos x=t^2 $
$ \therefore 2 t+\frac{t^2-1}{2}+2=0 \Rightarrow t^2+4 t+3=0$
$\Rightarrow t=-1,-3 \Rightarrow \sin x+\cos x=-1$
$ \Rightarrow \cos \left(x-\frac{\pi}{4}\right)=\frac{-1}{\sqrt{2}}=\cos \frac{3 \pi}{4}$
$\Rightarrow x-\frac{\pi}{4}=2 n \pi \pm \frac{3 \pi}{4}$
$ \Rightarrow x=2 n \pi+\pi, 2 n \pi-\frac{\pi}{2}$
$\Rightarrow x=2 n \pi+\pi$ at which $\text{cosec} x$ is not defined
$\therefore x=2 n \pi-\frac{\pi}{2}$.