Question

Describe, in your own words, the identity sin^2(x) + cos^2(x) = 1.

Answer 1

Look at the attached figure. We define the unit circle as a circle with center $A$ in the origin (0,0) and radius 1.

Then, we consider a point P on the circumference. We call $\alpha$ the angle between the positive half of the x axis and the radius AP.

We define

$\cos(\alpha) = \overline{AD},\quad \sin(\alpha) = \overline{AC}$

As you can see, ACD is a right triangle, and so we have

$\overline{AD}^2+\overline{AC}^2=\overline{AP}^2$

But since we know that AD is the cosine, AC is the sine, and AP is the radius (which is 1, and remains 1 when squared), we have just found out that

$\cos(\alpha)^2+\sin(\alpha)^2=1$