Common trig identities
The formulas you will use most often, organized by what they do.
Reciprocal identities
Each reciprocal function is one divided by its partner.
Quotient identities
Tangent and cotangent can always be rewritten using sine and cosine.
Pythagorean identities
The first identity generates the other two by division.
Even and odd identities
Cosine and secant are even; the other four trig functions are odd.
Cofunction identities
Complementary angles swap each function with its cofunction.
Sum and difference identities
The sine sign stays the same. The cosine sign switches. Each case is written separately for clarity.
Double-angle identities
The three cosine forms are equivalent; choose the one that fits what you know.
Half-angle identities
Choose ± from the quadrant containing the half-angle—not from the quadrant containing θ.
Power-reducing identities
These are rearranged double-angle identities and are especially useful in calculus.
Product-to-sum identities
Use these to turn products of trig functions into sums or differences.
Sum-to-product identities
Use the average of the angles and half their difference.
Do not memorize every line independently.
Start with the reciprocal, quotient, and first Pythagorean identities. Then learn the patterns in the angle formulas. Most of the longer identities become much easier to reconstruct once those foundations are automatic.