To prove this Pythagorean Identity, we have to know that:
cot = 1 / tan
and we also know that:
tan = sin / cos
Therefore,
cot = 1 / tan = cos / sin
So we can write the given equation in the form of:
1 + (cos^2 θ / sin^2 θ) = csc^2 θ
Expanding the left hand side of the equation:
(sin^2 θ / sin^2 θ) + (cos^2 θ / sin^2 θ) = csc^2 θ
(sin^2 θ + cos^2 θ) / sin^2 θ = csc^2 θ
We know that given a unit circle, sin^2 θ + cos^2 θ = 1. So:
1 / sin^2 θ = csc^2 θ
The equation above is already true basing on the
trigonometric identities. Therefore:
csc^2 θ = csc^2 θ
