(2cos^2x-1)(1+tan^2x)=1-tan^2x Please Help?How to prove the identity

First of all, we'll use the Pythagorean identity to
re-write the second factor of the product:

1 + x
= 1/ x

The left side will
become:

(2x - 1 )*(1/
x)

Now, we'll remove the brackets from the left and we'll
re-write the identity to be demonstrated:

2 - 1/ x
= 1 - x

We'll use again the Pythagorean identity
to re-write the second term of the difference from the right
side:

1 + x = 1/ x => x
= 1/ x - 1

The identity to be verified will
become:

2 - 1/ x = 1 - (1/x - 1
)

We'll remove the brackets from the right
side:

2 - 1/ x = 1 - 1/ x +
1

We'll combine like terms and we'll
get:

2 - 1/ x =  2 - 1/
x

Since LHS = RHS, therefore the given
identity (2 x - 1)(1+x ) = 1 - x is
verified.