How do you prove that $-2cosbeta(sin alpha - cosbeta) = (sin alpha - cosbeta)^2 + cos^2beta - sin^2alpha$ ?

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Answer 1 · 2017-02-15T20:02:36

You're going to want to expand everything on the right.

$RHS$ :

$(sin alpha - cos beta)(sin alpha - cos beta) + (cos beta + sin alpha)(cos beta - sin alpha)$

$sin^2alpha - 2cosbetasinalpha + cos^2beta + cos^2beta -sin^2alpha$

$2cos^2beta - 2cosbetasinalpha$

$2cosbeta(cos beta - sin alpha)$

$-2cosbeta(sin alpha - cos beta)$

The left hand side now equals the right hand side, so the identity has been proved.

Hopefully this helps!

How do you prove that -2cosbeta(sin alpha - cosbeta) = (sin alpha - cosbeta)^2 + cos^2beta - sin^2alpha-2cosbeta(sin alpha - cosbeta) = (sin alpha - cosbeta)^2 + cos^2beta - sin^2alpha?