The identity to be proved is sin 2x = (tan x)(1 + cos 2x)

Let's start from the right hand side

(tan x)(1 + cos 2x)

use tan x = sin x / cos x and cos 2x = 2(cos x)^2 - 1

=> (sin x / cos x)(1 + 2*(cos x)^2 - 1)

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

=> 2*sin x*cos x

=> sin 2x

which is the left hand side

**This proves the identity is sin 2x = (tan x)(1 + cos 2x)**

We know that 1 + cos 2x = 2 (cos x)^2 (half angle identity)

Also, we know that the tangent function is:

tan x = sin x/cos x

We'll manage the right side of the equality:

(tan x)(1 + cos 2x) = (sin x)*2 (cos x)^2/(cos x)

We'll simpliy and we'll get:

RHS = (tan x)(1 + cos 2x) = 2sin x*cos x = LHS

We notice that we've get the expression from the left side, since 2sin x*cos x = sin 2x

**The given identity sin 2x = (tan x)(1 + cos 2x) is verified.**