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