Prove that sin^4 x - cos^4x = 2sin^2x - 1
2 Answers
justaguide | College Teacher | (Level 2) Distinguished Educator
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`
tonys538 | Student, Undergraduate | (Level 1) Valedictorian
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