We have to prove: sin^4(theta) - cos^4(theta) = sin^2(theta) - cos^2(theta)
First let's write the terms in a standard form and use x instead of theta.
So we have to prove (sin x)^4 - (cos x)^4 = (sin x)^2 - (cos x)^2
Start with the left hand side:
(sin x)^4...
See This Answer Now
Start your subscription to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.
Already a member? Log in here.
We have to prove: sin^4(theta) - cos^4(theta) = sin^2(theta) - cos^2(theta)
First let's write the terms in a standard form and use x instead of theta.
So we have to prove (sin x)^4 - (cos x)^4 = (sin x)^2 - (cos x)^2
Start with the left hand side:
(sin x)^4 - (cos x)^4
we use the relation x^2 - y^2 = (x - y)(x + y)
=> [(sin x)^2 - (cos x)^2][(sin x)^2 + (cos x)^2]
we know that [(sin x)^2 + (cos x)^2] = 1
=> [(sin x)^2 - (cos x)^2] * 1
=> [(sin x)^2 - (cos x)^2]
which is the right hand side.
This proves that sin^4(theta) - cos^4(theta) = sin^2(theta) - cos^2(theta)