Prove that tanx=2sinx*cosx/(1+cos^2x-sin^2x) .

We'll replace 1 from denominator of the fraction by
Pythagorean identity:

(sin x)^2 + (cos x)^2 =
1

The denominator will
become:

(sin x)^2 + (cos x)^2 + (cos x)^2 - (sin x)^2 = 2
(cos x)^2

We'll re-write the
fraction:

2sin x*cos x/2 (cos x)^2 = sin x/cos
x

But the fraction sin x/cos x represents the tangent
function.

Since the LHS = RHS, then the given
identity tan x = 2 sinx*cos x/[1 + (cos x)^2 - (sin x)^2] is
true.