If one examines some of the above functions carefully, the sin and cosine functions are actually the coordinates of a point on the unit circle. This implies that the most important fundamental formula in trigonometry is as follows— one that some people call the “magic identity” but is more commonly known as one of the Pythagorean identities:

cos²θ + sin²θ = 1, in which θ is any real number.

This identity can be used in the following step-by-step example:

Show that:

sec²θ = tan²θ + 1

Because secθ = 1/cosθ (from the reciprocal identities), and tanθ = sinθ/cosθ (from the ratio or quotient identities) then:

tan² + 1 =

(sinθ/cosθ²) + 1 =

(sin²θ/cos²θ) + 1 =

(sin²θ + cos²θ)/cos²θ (where cos²θ divided by itself equals 1).

Then (using the Pythagorean identity):

1/cos²θ= sec²θ (from the reciprocal identity).