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

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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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tonys538 | Student, Undergraduate | (Level 1) Valedictorian

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To prove the identity sin^4 x - cos^4x = 2sin^2x - 1, use the trigonometric relation sin^2x + cos^2x = 1.

2*sin^2x - 1

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

= 2*sin^2x - sin^2x - cos^2x

= sin^2x - cos^2x

This can be multiplied by cos^2x + sin^2x the result of multiplying a number x by 1 is the number x.

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

= sin^4x - cos^4x

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