Establish the identity a cos2 theta 12cos4 theta sin4 thet

Establish the identity. (a cos^2 theta - 1)^2/cos^4 theta - sin^4 theta = 1- 2 sin^2 theta Use a Pythagorean identity to rewrite the numerator in terms of cos theta and sin theta and factor the denominator into two factors by factoring the difference of two squares. After cancelling common factors from the previous fraction, use a Pythagorean identity to rewrite the denominator Write the new denominator below. The entire expression can now be rewritten as 1 -2 sin^theta using what? Pythagorean Identity

Solution

I am taking Theta as A
We know that cos² A + sin²A = 1
Also we know that X² - Y² = (X+Y)(X-Y)

Now
( 2cos²A - 1 )² / cos⁴A - sin⁴A
=> [ 2cos²A - (cos²A + sin²A) ]² / (cos²A)² - (sin²A)²
=> ( cos²A - sin²A )² / (cos²A + sin²A)(cos²A - sin²A)

Cancelling (cos²A - sin²A) in the numerator and the denominator.
=> (cos²A - sin²A) / (cos²A + sin²A)
We know (cos²A + sin²A) = 1

=> cos²A - sin²A
=> (1 - sin²A) - sin²A
=> 1 - 2sin²A
Hence Proved