Prove that Cos^4A-Sin^4A+1=2Cos^2A

1.	Prove that Cos^4A-Sin^4A+1=2Cos^2A
Answer» 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.

Answer»

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.

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