cotx sec^4x = cotx + 2tanx + tan^3x
Taking left hand side,
= cotx + 2tanx + tan^3x
= (cosx / sinx) + (2sinx / cosx) + (sin^3x / cos^3x)
Taking L.C.M;
= (cos^4x + 2sin^2x cos^2x + sin^4x) / sinx cos^3x
Using formula a^2 + b^2 + 2ab = (a + b)^2
= (cos^2x + sin^2x)^2 / sinx cos^3x
As cos^2x + sin^2x = 1
= 1 / sinx cos^3x
= cosx / sinx cos^4x
= (cos x / sinx) * (1/cos^4x)
= cotx sec^4x
Hence right hand and left side are equal.
