TAN2X(1+COT2X)=1/1-SIN2X

Answer 1

as written, it is clearly not an identity, since when 2x = π/4, we have
1(1+1) = 1/(1 - 1/√2)
which is not true.

Care to fix your typos?

Answer 2

If your question means:

Prove

tan² x ( 1 + cot² x ) = 1/ ( 1 - sin² x )

then you should use identities:

1 - sin² x = cos² x

tan x = sin x / cos x

tan² x = sin² x / cos² x

tan x • cot x = 1

tan² x • cot² x = 1

Now

tan² x ( 1 + cot² x ) = 1/ ( 1 - sin² x )

can be write as:

tan² x ( 1 + cot² x ) = 1 / cos² x

tan² x • 1 + tan² x • cot² x = 1 / cos² x

tan² x + 1 = 1 / cos² x

sin² x / cos² x + 1 = 1 / cos² x

Multiply both sides by cos² x

sin² x + cos² x = 1

This is one of the basic trigonometric identities, which is of course true.

So:

tan² x ( 1 + cot² x ) = 1/ ( 1 - sin² x )

is also true.