Question:
For all real values of $x, \cot x-2 \cot 2 x$ is equal to
(a) $\tan 2 x$
(b) $\tan x$
(c) $-\cot 3 x$
(d) none of these
Solution:
(b) $\tan x$
We have,
$\cot x-2 \cot 2 x=\cot x-2 \frac{\cot ^{2} x-1}{2 \cot x}$
$=\frac{\cot ^{2} x-\cot ^{2} x+1}{\cot x}$
$=\frac{1}{\cot x}$
$=\tan x$