Question

If $\cot \theta = \sin 2\theta $ (where $\theta \ne n\pi $, $n$ is an integer), $\theta = $
(A) ${45^\circ }$ and ${60^\circ }$
(B) ${45^\circ }$ and ${90^\circ }$
(C) only ${45^\circ }$
(D) only ${90^\circ }$

Answer 1

Hint:
While solving this question, first of all write $\cot \theta $ in terms of $\sin \theta $ and $\cos \theta $ and use formula $\sin 2\theta = 2\sin \theta \cos \theta $ to solve the equation using the mathematical operations as per requirement.

Complete step by step solution:
Given, $\cot \theta = \sin 2\theta $ ….. (1)
Using $\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}$ and $\sin 2\theta = 2\sin \theta \cos \theta $, equation (1) becomes
$ \Rightarrow $$\dfrac{{\cos \theta }}{{\sin \theta }} = 2\sin \theta \cos \theta $
$ \Rightarrow \cos \theta = 2{\sin ^2}\theta \cos \theta $
$ \Rightarrow \cos \theta - 2{\sin ^2}\theta \cos \theta = 0$
$ \Rightarrow \cos \theta \left( {1 - 2{{\sin }^2}\theta } \right) = 0$
But $1 - 2{\sin ^2}\theta = \cos 2\theta \,$
$ \Rightarrow \cos \theta \left( {\cos 2\theta } \right) = 0$
$ \Rightarrow \cos \theta = 0$ or $\cos 2\theta = 0$
$ \Rightarrow \cos \theta = \cos {90^\circ }$ or $\cos 2\theta = \cos {90^\circ }$ $\left( {\because \cos {{90}^\circ } = 0} \right)$
$ \Rightarrow \theta = {90^\circ }$ or $\theta = {45^\circ }$
$\therefore \theta $ can be ${45^\circ }$ or ${90^\circ }$

Hence, option (B) is the correct answer.

Note:
Here it may be noted that $\cos 2\theta $ has three formulas, i.e., \[\cos 2\theta = 2{\cos ^2}\theta - 1 = 1 - 2{\sin ^2}\theta \]. So use the formula of $\cos 2\theta $ as per requirement. For ex- In above question, use $1 - 2{\sin ^2}\theta = \cos 2\theta \,$.