Question

Prove that (cos 2α - cos 2β) / (sin 2α + sin 2β) = tan(β - α)
Note: I don't mean cos square alpha. It is cos 2 alpha as it is written...

Answer 1

$\frac{cos2 \alpha - cos2 \beta }{sin2 \alpha + sin2 \beta } \\ = \frac{2sin( \frac{2 \alpha + 2 \beta }{2} )sin( \frac{2 \beta - 2 \alpha }{2}) }{2sin( \frac{2 \alpha + 2 \beta }{2} )cos( \frac{2 \alpha - 2 \beta }{2}) } \\ = \frac{sin( \beta - \alpha) }{cos( \alpha - \beta )} \\ = \frac{sin( \beta - \alpha )}{cos( \beta - \alpha )} \\ = tan( \beta - \alpha )$

since cos(- θ)=cosθ