Question

Verify each trigonometric equation by substituting identities to match the right hand side of the equation to the left hand side of the equation.
cot x sec4x = cot x + 2 tan x + tan3x

Answer 1

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.