Question

Prove: $2\ tan^{-1}(cos\ x)=tan^{-1}(2\ cosec\ x)$

Answer 1

2 tan-1(cos x) =tan-1(2 cosec x)
$\implies$tan-1$\frac {2\ cos\ x}{1-cos^2\ x}$ = tan-1(2 cosec x) [2tan-1x = tan-1$\frac {2x}{1-x^2}$]
\implies$$\frac {2\ cos\ x}{1-cos^2\ x} = 2 cosec x
\implies$$\frac {2\ cos\ x}{sin^2\ x} = $\frac {2}{sin\ x}$
$\implies$cos x = sin x
$\implies$tan x=1
So, x = $\frac {\pi}{4}$