Question

How do you verify `(1 + cos theta)(1 - cos theta) = sin^2 theta`?

Answer 1

Use Pythagoras and the basic definitions of sine and cos
sin x = `o/h`
cos x = `a/h`
where o =the side opposite the angle x, a = the side adjacent to the angle x and h equals the hypotenuse of the right-angled triangle.

Pythagora states `h^2 = a^2 + o^2`
therefore `1 = a^2/h^2 + o^2/h^2`
so 1 = `cos^2(x) + sin^2 (x)`
`sin^2(x) = 1 - cos^2(x)`
tthen `sin^2(x) = (1 - cos(x))(1+cos(x))`

Answer 2

`(1+cos theta)(1-cos theta) = 1-cos^2 theta`

`= (sin^2 theta + cos^2 theta )- cos^2 theta`

`= sin^2 theta + (cos^2 theta - cos^2 theta)`

`= sin^2 theta`