Question

Explain in terms of the unit circle and geometry why the trig identity cos2 x + sin2 x = 1 holds true for all values of x.

Answer 1

We can imagine the given equation cos2 x + sin2 x = 1 to be in the form of the hypotenuse equation.

Where cos x and sin x is the base and height of the triangle and 1 is the hypotenuse.

Since 1 is the hypotenuse, it also indicates to the radius of the circle. Supposing that the center of the circle is at the origin, then at all values of x (at all values of the angle), the radius of the circle will always be the same which is equal to 1.

Therefore the equation cos2 x + sin2 x = 1 holds to be true for all values of x for a unit circle.