Question
For all real values of $$\theta ,\cot \theta - 2\cot 2\theta $$ is equal to
A.
$$\tan 2\theta $$
B.
$$\tan \theta $$
C.
$$ - \cot 3\theta $$
D.
None of these
Answer :
$$\tan \theta $$
Solution :
$$\eqalign{
& \cot \,\theta - 2\,\cot \,2\theta \cr
& = \frac{{\cos \,\theta }}{{\sin \,\theta }} - \frac{{2\,\cos \,2\theta }}{{\sin \,2\theta }} \cr
& = \frac{{\sin \,\theta .\cos \,\theta - 2\,\cos \,2\theta .\sin \,\theta }}{{\sin \,\theta .\sin \,2\theta }} \cr
& = \frac{{\sin \left( {2\theta - \theta } \right) - \cos \,2\theta .\sin \,\theta }}{{\sin \,\theta .\sin \,2\theta }} \cr
& = \frac{{\sin \,\theta \left\{ {1 - 1 + 2\,{{\sin }^2}\theta } \right\}}}{{\sin \,\theta .2\,\sin \,\theta .\cos \,\theta }} \cr
& = \tan \,\theta \cr
& {\text{Hence, option B is the correct option}}{\text{.}} \cr} $$