Question

State true or false.

$\frac{\cos \theta +\cos \varphi }{\sin \theta +\sin \varphi }=\cot \left(\frac{\theta +\varphi }{2}\right)$.

• A. True
• B. False

Answer 1

The correct option is A True

As we know that

$\cos a+\cos b=2\cos \frac{a+b}{2}\cos \frac{a-b}{2}$

$\sin a+\sin b=2\sin \frac{a+b}{2}\cos \frac{a-b}{2}$

$⟹\frac{\cos \theta +\cos \varphi }{\sin \theta +\sin \varphi }=\frac{2\cos \frac{\theta +\varphi }{2}\cos \frac{\theta -\varphi }{2}}{2\sin \frac{\theta +\varphi }{2}\cos \frac{\theta -\varphi }{2}}=\frac{\cos \frac{\theta +\varphi }{2}}{\sin \frac{\theta +\varphi }{2}}=\cot (\frac{\theta +\varphi }{2})$

The relation is $\text{True}$