Trig Identities
This is my reference page for common trigonometric identities and formulas.
Pythagorean Identities:
Definitions:
Half-Angle Identities:
Double-Angle Identities:
Period & Symmetry:
Hyperbolic Functions
Definitions:
Pythagorean Identities:
Half-Angle Identities:
Double-Angle Identities:
Symmetry:
Derivations
Double Angle
The double angle formulas can be derived in just one step by using a rotation matrix:
Multiplying this matrix by a point (represented as a column vector) rotates the point by the angle θ. If you start with the (x,y) coordinates of the trig triangle's point on the circle, which has already been rotated by θ, the result will be a point that has been rotated by θ twice, which is 2θ.
The half-angle formulas can be derived by starting with half the angle, then substituting the double angle formulas, and solving for θ.
Angle Sum
The angle sum formulas can be derived by rotating by an arbitrary angle ϕ:
Symmetry
The symmetry formulas can be derived by the inverse matrix, which rotates by -θ:
Hyperbolic
The hyperbolic formulas can be derived by substituting the complex* definitions of the hyperbolic functions and using the regular Euclidean trig formulas. For example:
* If you're unfamiliar with complex numbers, just treat i as an unknown variable, but then replace every i2 with -1, if it ever appears.