The identity `cos^4 x + 1 - sin^4 x = 2cos^2 x` has to be proved.

`cos^4 x + 1 - sin^4 x`

=> `(1 - sin^2 x)^2 + 1 - sin^4 x`

=> `1 + sin^4 x - 2*sin^2x + 1 - sin^4 x`

=> `2 - 2*sin^2 x`

=>...

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The identity `cos^4 x + 1 - sin^4 x = 2cos^2 x` has to be proved.

`cos^4 x + 1 - sin^4 x`

=> `(1 - sin^2 x)^2 + 1 - sin^4 x`

=> `1 + sin^4 x - 2*sin^2x + 1 - sin^4 x`

=> `2 - 2*sin^2 x`

=> `2(1 - sin^2x)`

=> `2*cos^2x`

**This proves that **`cos^4 x + 1 - sin^4 x = 2cos^2 x `