Use the Pythagorean identity to rewrite the tangent:
[tex]\dfrac2{1+\tan^2x}=\dfrac2{\sec^2x}=2\cos^2x[/tex]
Then use the double angle identity for cosine:
[tex]\cos^2x=\dfrac{1+\cos(2x)}2[/tex]
[tex]\implies2\cos^2x=1+\cos(2x)[/tex]
and the identity is proved.
