Question

How do you prove `sin (x+y)sin(x-y) = cos^2y-cos^2x`?

Answer 1

You should use the following trigonometrical identities:

`sin (x + y) = sin x cos y + cos x sin y`

`sin (x - y) = sin x cos y - cos x sin y`

This will lead you to:

`sin(x + y )sin (x - y)`

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

`= (color(blue)(sin x cos y) + color(red)(cos x sin y))(color(blue)(sin x cos y) - color(red)(cos x sin y))`

... use the formula `(a+b)(a-b) = a^2 - b^2`:

`= sin^2 x cos^2 y - cos^2 x sin^2 y`

... use `sin^2 x + cos^2 x = 1 " " <=> " " sin^2 x = 1 - cos^2 x`:

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

`= cos^2 y - cos^2 x cos^2 y - cos^2 x + cos^2 x cos^2 y`

`= cos^2 y - cancel(cos^2 x cos^2 y) - cos^2 x + cancel(cos^2 x cos^2 y)`

`= cos^2 y - cos^2 x`