Question

$\cot \left(\frac{x}{2}\right)-\csc \left(\frac{x}{2}\right)=\cot x$, then x belongs to

Answer 1

$\cot \left(\frac{x}{2}\right)-\csc \left(\frac{x}{2}\right)=\frac{\cos x}{\sin x}$

$\frac{\cos \left(\frac{x}{2}\right)-1}{\sin \left(\frac{x}{2}\right)}=\frac{\cos x}{\sin x}=\frac{\cos x}{2\sin \left(\frac{x}{2}\right)\cos \left(\frac{x}{2}\right)}$

$2\cos \left(\frac{x}{2}\right)\left(\cos \right(\frac{x}{2})-1)=\cos x$

$2{\cos }^2\left(\frac{x}{2}\right)-2\cos \left(\frac{x}{2}\right)=\cos x$

$1+\cos x-2\cos \left(\frac{x}{2}\right)=\cos x$

$\cos \left(\frac{x}{2}\right)=\frac{1}{2}$

$\frac{x}{2}=2n\pi \pm \frac{\pi }{3}$

$x=4n\pi \pm \frac{2\pi }{3}$