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