Question

If f(2tanx/(1 + tan²x)) = 1/2(1 + cos2x)(sec²x + 2tanx) then find f(x).

Answer 1

We have f(2tanx/(1 + tan²x))

= 1/2(1 + cos2x)(sec²x + tanx)

= 1/2x 2cos2x x (1 + tan²x + 2tanx)

= cos²x x (1 + tanx)²

= {cosx x (1 + tanx)}²

= (cosx + sinx)²

= 1 + sin(2x)

Thus, f(sin 2x) = 1 + sin (2x)

⇒ f(x) = 1 + x