Can we prove that (sin x)^4 + (cos x)^4 = 1 is always true

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It has to be determined whether `sin^4 x + cos^4 x = 1` is an identity.

We know that `cos^2 x + sin^2 x = 1` is true for all values of x.

If we take the square of both the sides, we get:

`[cos^2x +sin^2x]^2 = 1^2`

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

Now the given relation is true only when `2*cos^2x*sin^2x = 0`

=>  `(2*sin 2x *cos 2x)^2/2 = 0`

=> `(sin4x)/2 = 0`

=> `sin 4x = 0`

=> `4x = 0 or 4x = 180`

=> `x = 0^@ or x = 45^@`

The given relation is not an identity as it is true for only a few values of x and does not hold for all values of x.

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