Prove the identity 12cotx tanxcsc2xSolution12 cot x tan x

Prove the identity

1/2(cotx + tanx)=csc2x

Solution

1/2 ( cot x + tan x ) = csc2x

proving left hand side of the identity

cot x = cos x / sin x

tan x = sin x / cos x

1/2 ( cos x / sin x + sin x / cos x)

1/2 ( cos^2 x + sin^2 x / cos x sin x )

= 1/2 ( 1 / cos x sin x)

= 1/ 2 sin x cos x = csc 2x

hence proved