Prove that sin^4 x - cos^4x = 2sin^2x - 1

Expert Answers

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

Start from the left hand side.

`sin^4x - cos^4x`

Use the relation `x^2 - y^2 = (x - y)(x +y)`

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

Use the property `sin^2x + cos^2x = 1`

= `(sin^2x - cos^2x)`

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

= `sin^2x - 1 + sin^2x`

= `2*sin^2x - 1`

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

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