Question

How do you verify `sin^2 x + sin^2(pi/2 -x) = 1`?

Answer 1

On the trig unit circle:

`sin ( pi/2 - x) = cos x`, then

`sin^2 x + sin^2 (pi/2 - x) = sin^2 x + cos^2 x = 1`

Answer 2

If `0 < x < pi/2` then another way to picture this is as follows:

Construct a right angled triangle with angles `x`, `(pi/2 - x)` and `pi/2`, with hypotenuse of length `1`. It is possible to do this since these angles add up to `pi`.

Then the side opposite the angle `x` will have length `sin x` and the side opposite the angle `(pi/2 - x)` will have length `sin (pi/2 - x)`.

By Pythagoras theorem, the sum of the squares of the lengths of these sides is equal to the square of the length of the hypotenuse.

So `sin^2 x + sin^2 (pi/2 - x) = 1^2 = 1`