Show that sin 2x=2tan x/(1+tan^2x)?

We'll manage the LHS.

We'll
write the formula that writes the sine of double angle in terms of sine and cosine of
the angle.

sin 2x = 2 sin x*cos
x

Now, we'll multiply and divide by cos x the right side,
to create the tangent function:

sin 2x = 2 sin x*cos x*cos
x/cos x

sin 2x = 2 tan x*`cos^(2)`
x

But, from Pythagorean identity, we'll
have:

1 + `tan^(2)` x = 1/`cos^(2)` x => `cos^(2)` x
= 1/(1+`tan^(2)` x)

sin 2x = 2 tan x/(1 + `tan^(2)`
x)

Therefore, the identity sin 2x = 2 tan
x/(1 + `tan^(2)` x) is verified.