# Prove that:- 1+ cos^2 2x= 2cos^4 x +2sin^4 x

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We know that cos(2x) = cos^2(x) - sin^2(x).

Then by the formula (a-b)^2 = a^2 + b^2 - 2ab we obtain

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

So our equality is equivalent to

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

which in turn is equivalent to

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

Here we recognize the formula (a+b)^2 = a^2 + b^2 + 2ab:

1 = (cos^2(x) + sin^2(x))^2.

But cos^2(x) + sin^2(x) always =1, and 1^2 also =1. So the proof is complete.

cos^4(x)+sin^4(x)= (cos^2(x)+sin^2(x))^2 - 2cos^2(x)sin^2(x)

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

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

2cos^4(x)+2sin^4(x)= 1 + cos^2(2x)

Proof done.