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

Expert Answers
lemjay eNotes educator| Certified Educator


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.

user1450001 | Student

Let x=sin(A)^2 and y=cos(A)^2, then

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

oldnick | Student
cos^4 A - sin^4 A +1 = (cos^2 A + sin^2 A)(cos^2 A - sin^2 A) +1 since sin^2 A + cos^2 A = 1 we get: cos^2 A - sen^2 A + 1 = cos^2 A -( 1-cos^2 A )+1 = cos^2 A -1+ cos^2 A +1 = 2cos^2 A