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 = \(\frac {\pi}{4}\)