Question

Prove that Cos^4A-Sin^4A+1=2Cos^2A​

Answer 1

Answer:

I learned this from a friend. Hope it helps

Step-by-step explanation:

First, group the first two terms and factor it.

(cos^4A-sin^4A)+1=2cos^2A

(cos^2A-sin^2A)(cos^2A+sin^2A)+1=2cos^2A

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

 

(cos^2A-sin^2A)(1)+1=2cos^2A

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

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

cos^2A+(-sin^2A+1)=2cos^2A

cos^2A+(1-sin^2A)=2cos^2A

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

cos^2A+cos^2A=2cos^2A

2cos^2A=2cos^2A

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.