Question

If cot(theta) +cot(pi/4+theta) =2, then the general value of theta​

Answer 1

Given:

If cot(theta) +cot(pi/4+theta) =2.

To Find:

Value of theta?

Solution:

it is given that-

cot(theta) +cot(pi/4+theta) =2

on solving this , we get

$\dfrac{{\sin \left( {\dfrac{\pi }{4} + \theta + \theta } \right)}}{{\sin \theta .\sin \left( {\dfrac{\pi }{4} + \theta } \right)}} = 2 \hfill$

$\dfrac{{\sin \dfrac{\pi }{4}\cos 2\theta + \cos \dfrac{\pi }{4}\sin 2\theta }}{{\sin \dfrac{\pi }{4}\cos \theta + \cos \dfrac{\pi }{4}\sin \theta }} = 2\sin \theta \hfill$

$\dfrac{{\cos 2\theta + \sin 2\theta }}{{\cos \theta + \sin \theta }} = 2\sin \theta \hfill \\\\ \cos 2\theta + \sin 2\theta = 2\sin \theta \cos \theta + 2{\sin ^2}\theta \hfill$

$(1 - 2{\sin ^2}\theta ) + \sin 2\theta = \sin 2\theta + 2{\sin ^2}\theta \hfill \\\\ 1 = 4{\sin ^2}\theta \hfill \\\\ \sin \theta = \frac{1}{2} = \sin \frac{\pi }{6} \hfill \\\\ \theta = \frac{\pi }{6} \hfill$