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`

Answer 2

Answer: Here is a copy paste-able, Edg2021 friendly version of the above answer^ :)

I defined the unit circle as a circle with center A in the origin (0,0) and radius 1. Then, I considered a point P on the circumference. I call a the angle between the positive half of the x axis and the radius AP. I defined that

cos(a)= line AD, sin(a)= line AC

ACD is a right triangle, and so,

line AD^2+ line AC^2 = line AP^2

But since I knew that AD is the cosine, AC is the sine, and AP is the radius (which is 1, and remains 1 when squared), I found that

cos(a)^2 + sin(a)^2 = 1