Question

Sin^4a+cos^4a=1+cos^2(2a)/2

Answer 1

(1-cos2a)²/4+(1+cos2a)²/4=(1+cos²2a)/2
1/4*(1-2cos2a+cos²2a+1+2cos2a+cos²2a)=(1+cos²2a)/2
1/4*(2+2cos²2a)=(1+cos²2a)/2
2*1/4*(1+cos²2a)=(1+cos²2a)/2
(1+cos²2a)/2=(1+cos²2a)/2

Answer 2

The trigonometric identity you've provided is incorrect. The correct identity involving sin and cos is:

$\sin^4(a) + \cos^4(a) = 1 - \frac{\cos^2(2a)}{2}$

This identity can be derived using various trigonometric identities and algebraic manipulations. If you'd like, I can provide a step-by-step derivation of this identity for you. Just let me know!