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

2 Answers
May 15, 2015

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#

May 17, 2015

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#