Reciprocal Identities

sin(x) == 1/csc(x)    <=>    csc(x) == 1/sin(x)

cos(x) == 1/sec(x)    <=>    sec(x) == 1/cos(x)

tan(x) == 1/cot(x)    <=>    cot(x) == 1/tan(x)

Quotient Identities

tan(x) == sin(x)/cos(x) == sec(x)/csc(x)

cot(x) = =cos(x)/sin(x) == csc(x)/sec(x)

Pythagorean Identities

sin^2(x) + cos^2(x) == 1

tan^2(x) + 1 == sec^2(x)

1 + cot^2(x) == csc^2(x)

Learn to derive these from the ratios based on right triangles and learn to use them when manipulating equations.

Example Derivation:

Pythagorean Identity:     x^2 + y^2 == r^2

Divide both sides by r^2 :     (x/r)^2 + (y/r)^2 == (r/r)^2 == 1

Replace ratios with their representative trig functions:     sin^2(x) + cos^2(x) == 1