Question

Prove $1+cot\theta \le cot\left(\frac{\theta }{2}\right)$ for all $0<\theta <\pi$

Answer 1

We have to prove that

$1+\cot \theta ⩽\cot \left(\frac{\theta }{2}\right)forall0<\theta <\pi$

Now,

$L.H.S=1+\cot \theta$

$=1+\frac{\cos \theta }{\sin \theta }$

$=\frac{\sin \theta +\cos \theta }{\sin \theta }$

And,

$R.H.S=\cot \frac{\theta }{2}$

Hence,

$L.H.S⩽R.H.S$

So, the following question is proved.