I need help. (cos x - sin y)/(cos y - sin x) = (cos y + sin x)/(cos x + sin y)?

Question

With proving the identity.

Answer 1

above is true if
(cos x - sin y)(cos x + sin y) = (cos y + sin x)(cos y - sin x)
or cos ^2 x - sin ^2 y = cos^2 y - sin ^2 x

if cos^2 x + sin ^2 x = cos^2 y + sin ^2 y
it is true as both are same
however the above proof is not elegent

Answer 2

just open them
cos ^2 x - sin ^2 y = cos^2 y - sin ^2 x

if cos^2 x + sin ^2 x = cos^2 y + sin ^2 y solved

Answer 3

(cos x - sin y)/(cos y - sin x) = (cos y + sin x)/(cos x + sin y)

Multiply through
(cos x - sin y) (cos x + sin y) = (cos y + sin x) (cos y - sin x)

Use the (a+b)(a-b)=(a^2-b^2) identity
(cos^2 x - sin^2 y) = (cos^2 y- sin^2 x)

1=cos^2+sin^2
1-sin^2 = cos^2
(1-sin^2 x - sin^2 y) = (cos^2 y - sin^2 x)

1-cos^2 = sin^2
(1 - sin^2 x - (1 - sin^2 y)) = (cos^2 y - sin^2 x)

Bring the negative through
(1- sin^2 x - 1 + sin^2 y) = (cos^2 y - sin^2 x)

Cancel the 1s
(-sin^2 x + sin^2 y) = (cos^2 y - sin^2 x)

Then just reorder
(cos^2 y - sin^2 x) = (cos^2 y - sin^2 x)