Question

State whether the given statement is true or false

$1+\cot \theta \ge \cot \theta /2$ for $0<\theta <\pi$. Find $\theta$ when equality signs holds.If this is true enter 1, else enter 0.

• A. True
• B. False

Answer 1

The correct option is B False

Given, $1+\frac{\cos \theta }{\sin \theta }\ge \frac{\cos \frac{\theta }{2}}{\sin \frac{\theta }{2}}$
$\frac{\sin \theta +\cos \theta }{\sin \theta }\ge \frac{\cos \frac{\theta }{2}}{\sin \frac{\theta }{2}}$
$\frac{\sin \theta +\cos \theta }{2\sin \frac{\theta }{2}.\cos \frac{\theta }{2}}\ge \frac{\cos \frac{\theta }{2}}{\sin \frac{\theta }{2}}$
$\frac{\sin \theta +\cos \theta }{2\sin \frac{\theta }{2}.\cos \frac{\theta }{2}}-\frac{2{\cos }^2\frac{\theta }{2}}{2\sin \frac{\theta }{2}.\cos \frac{\theta }{2}}\ge 0$
$\frac{1}{\sin \theta }[\sin \theta +\cos \theta -(1+\cos \theta \left)\right]\ge 0$
$cosec\theta [sin\theta -1]\ge 0$
$1-cosec\theta \ge 0$
$1\ge cosec\theta$
$\sin \theta \ge 1$
$\sin \theta =1$ or $\sin \theta >1$
But, $|\sin \theta |\le 1$
Hence, $\sin \theta =1$
$\theta =\frac{\pi }{2}$.