Trigonometric Identities
A copy-able sheet of the reciprocal, Pythagorean, sum, double-angle, and half-angle identities.
Click any identity to copy the formula.
The core trig identities fall into a few families: reciprocal and quotient (csc θ = 1/sin θ, tan θ = sin θ/cos θ), the three Pythagorean identities (sin²θ + cos²θ = 1), even-odd and cofunction relations, and the sum, double-angle, half-angle, and product-to-sum formulas. Click any identity above to copy it.
How the families connect
Almost every identity traces back to two starting points: the definitions of the six functions and the single Pythagorean identity sin²θ + cos²θ = 1. The reciprocal identities (csc, sec, cot) and the quotient identities (tan = sin/cos) come straight from the definitions. Divide sin²θ + cos²θ = 1 through by cos²θ or sin²θ and you get the other two Pythagorean forms, 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ.
Angles you can combine
The sum and difference formulas let you split any combined angle, and everything else follows from them. Set A = B in sin(A + B) and cos(A + B) and you land on the double-angle identities; the half-angle identities are the double-angle formulas for cos 2θ solved for sin(θ/2) and cos(θ/2). The product-to-sum formulas rewrite a product of sines and cosines as a sum, which is handy for integration.
The three Pythagorean identities
All three come from sin²θ + cos²θ = 1 by dividing through by cos²θ or sin²θ.
| Identity | Derived by |
|---|---|
| sin²θ + cos²θ = 1 | the unit-circle definition |
| 1 + tan²θ = sec²θ | dividing the first by cos²θ |
| 1 + cot²θ = csc²θ | dividing the first by sin²θ |
Double-angle identities
Cosine has three equivalent forms — pick the one that matches what you already know.
| Function | Identity |
|---|---|
| sin 2θ | 2 sin θ cos θ |
| cos 2θ | cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ |
| tan 2θ | 2 tan θ / (1 − tan²θ) |
Sum and difference identities
Read the top sign with a sum and the bottom sign with a difference; note how cosine flips the middle sign.
| Function | Identity |
|---|---|
| sin(A ± B) | sin A cos B ± cos A sin B |
| cos(A ± B) | cos A cos B ∓ sin A sin B |
| tan(A ± B) | (tan A ± tan B) / (1 ∓ tan A tan B) |
Deriving one identity from another
You rarely need to memorise the whole sheet. The double-angle sine is just sin(A + B) with A = B: sin(θ + θ) = sin θ cos θ + cos θ sin θ = 2 sin θ cos θ. The half-angle cosine comes from cos 2θ = 2 cos²θ − 1: replace 2θ with θ, solve for the squared term, and take the root to get cos(θ/2) = ±√((1 + cos θ)/2). The sign depends on the quadrant of θ/2.
Where the signs matter
In the cosine sum formula the middle sign is opposite the angle sign — cos(A + B) uses a minus, cos(A − B) uses a plus. The ∓ notation captures that: it is the reverse of the ± in the same line. In the tangent formula the denominator sign flips the same way. Getting these signs right is the most common place trig work goes wrong.