Question

Sin^4 (x) + cos^4 (x) = cos2x +1/4

Answer 1

(1-сos2x)²/4+(1+cos2x)²/4-cos²2x-1/4=0

1-2cos2x+cos²2x+1+2cos2x+cos²2x-4cos²2x-1=0

1-2cos²2x=0

cos²2x=1/2

cos2x=-1/2⇒2x=+-2π/3+2πk⇒x=+-π/3+πk,k∈z

cos2x=1/2⇒2x=+-π/3+2πk⇒x=+-π/6+πk,k∈z

Answer 2

The equation you provided, sin^4(x) + cos^4(x) = cos(2x) + 1/4, is not correct. The true identity is slightly different. The correct trigonometric identity is:

sin^4(x) + cos^4(x) = 1 - 1/2 * sin^2(2x)

This identity can be derived using various trigonometric identities and properties. If you're interested in the derivation, I can provide it for you.