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Verify the given the trigonometric identities

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The trigonometry identities to be proved:

1) cos(4x) = 1 - 8sin^2(x) + 8sin^4(x)
2) tan(4x) = [4tan(x) - 4tan^3(x)]/[1 - 6tan^2(x) + tan^4(x)]
3) 2cos(4x)sin(2x) = 2sin(3x)cos(3x) - 2sin(x)cos(x)
4) 2sin(x)sin(2x) = 4cos(x)sin^2(x)
5) cos(x+y)cos(x-y) = cos^2(x) + cos^2(y) - 1
6) cos(x+y)sin(x-y) = sin(x)cos(x) - sin(y)cos(y)

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https://brainmass.com/math/trigonometry/verify-given-trigonometric-identities-472750

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In this posting, proofs of the given trigonometric identities are shown.

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