solve the following equation 2tan^-1(cosx) = tan^-1(2cosecx)

Here , 2tan^-1( cosx) = tan^-1( 2 cosec x)
Or tan^-1( 2cosx / 1 – cos²x) = tan^-1( 2 cosecx )   [ use 2 tan^-1 x = tan^-1 ( 2x/1 - x²) ]
Or 2 cosx/sin²x = 2/sinx
Or sinx = cosx
Or tanx = 1
Or x = π/4
Hope that helps.