Prove that `cos^4A - sin^4A +1 = 2cos^2A`

Expert Answers

An illustration of the letter 'A' in a speech bubbles


First, group the first two terms and factor it.



Then, apply the Pythagorean identity which is  `cos^2theta + sin^2theta=1` .



`cos^2A-sin^2A + 1=2cos^2A`

Then, group the second and last term at the left side of the equation.



To simplify the expression inside the parenthesis, apply the Pythagorean identity again.



Since left side simplifies to `2cos^2A` which is the same term with the right side, hence it proves that the given equation is an identity.

Approved by eNotes Editorial Team