The three Pythagorean identities
The fundamental Pythagorean identity states that sin squared theta plus cos squared theta equals one for every angle. Dividing both sides by cos squared gives one plus tan squared equals sec squared. Dividing by sin squared gives one plus cot squared equals csc squared.
sin^2(theta) + cos^2(theta) = 1
The fundamental Pythagorean identity, valid for all angles.
1 + tan^2(theta) = sec^2(theta)
Derived by dividing the fundamental identity by cos squared theta.
1 + cot^2(theta) = csc^2(theta)
Derived by dividing the fundamental identity by sin squared theta.