Prove that sin^4 x+cos^4 x+(sin^2 2x)/2=1

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We have to prove that (sin x)^4 + (cos x)^4 + (sin 2x)^2/2 = 1

(sin x)^4 + (cos x)^4 + (sin 2x)^2/2

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

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

=> (sin x)^4 + (cos x)^4 + 2*(sin x)^2*(cos x)^2

=> [(sin x)^2 + (cos x)^2]^2

=> 1^2

=> 1

This proves that (sin x)^4 + (cos x)^4 + (sin 2x)^2/2 = 1

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