Prove that 4(sin^4x + cos^4x) = (cos4x + 3)?

Question

Well done if you get this one it's hard
I think you have to start by 'pretending' that sin^4x + cos^4x is sin^2x + cos^2x and use sin^2x + cos^2x = 1
Then you have to do something with double angle formulae maybe... I don't know!!!

Answer 1

Prove the identity.

4(sin^4 x + cos^4 x) = (cos 4x + 3)
_________

Left Hand Side = 4(sin^4 x + cos^4 x)

= 4sin^4 x + 4cos^4 x = (2sin²x)² + (2cos²x)²

= (1 - cos 2x)² + (1 + cos 2x)²

= (1 - 2cos 2x + cos²2x) + (1 + 2cos 2x + cos²2x)

= 2 + 2cos²2x = 2 + (1 + cos 4x)

= 3 + cos 4x = Right Hand Side

Answer 2

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