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Which trig identity should I use?
Look at what is in front of you, not at a list to memorise. The five shapes below cover almost every homework problem in this area: a squared term, a sum inside a function, a doubled or halved angle, a product of two different functions, or a sum of the same function.
How to decide, shape by shape
Each row is generated from the same rules the tool above uses, so the two cannot disagree.
When your expression has squared terms
A squared sine or cosine is the single loudest signal in trigonometry. Either sin²θ + cos²θ = 1 kills the pair outright, or a power-reducing formula rewrites one of them without any power.
- sin²θ + cos²θ = 1 — Pythagorean identity · proof
- sin²x = (1 − cos 2x)/2 — Power-reducing formula for sin² · proof
- 1 + tan²θ = sec²θ — Pythagorean identity (tangent form) · proof
- cos²x = (1 + cos 2x)/2 — Power-reducing formula for cos² · proof
- 1 + cot²θ = csc²θ — Pythagorean identity (cotangent form) · proof
- cos 2x = 1 − 2 sin²x — Double-angle identity for cosine (form 3) · proof
When you see 1 + (something)²
1 + tan²θ, 1 + cot²θ and their csc/sec partners come straight from the Pythagorean family. Under a square root — √(1 + tan²θ) — this is usually the whole problem.
- 1 + tan²θ = sec²θ — Pythagorean identity (tangent form) · proof
- 1 + cot²θ = csc²θ — Pythagorean identity (cotangent form) · proof
When one angle is a sum or difference
sin(a + b), cos(a − b): expand when you need the pieces, collapse when you recognise the expansion. This is also how you compute sin 15° exactly.
- sin(A + B) = sin A cos B + cos A sin B — Sine of a sum · proof
- sin(A − B) = sin A cos B − cos A sin B — Sine of a difference · proof
- cos(A + B) = cos A cos B − sin A sin B — Cosine of a sum · proof
- cos(A − B) = cos A cos B + sin A sin B — Cosine of a difference · proof
- tan(A + B) = (tan A + tan B) / (1 − tan A tan B) — Tangent of a sum · proof
- tan(A − B) = (tan A − tan B) / (1 + tan A tan B) — Tangent of a difference · proof
- tan A + tan B = sin(A + B) / (cos A cos B) — Tangent sum as a single fraction · proof
When the angle is doubled and you want the single angle
sin 2x, cos 2x, tan 2x in the question but x in the answers means: rewrite in terms of x. Cosine's three forms are the choice to make.
- sin 2x = 2 sin x cos x — Double-angle identity for sine · proof
- cos 2x = cos²x − sin²x — Double-angle identity for cosine (form 1) · proof
- cos 2x = 2 cos²x − 1 — Double-angle identity for cosine (form 2) · proof
- cos 2x = 1 − 2 sin²x — Double-angle identity for cosine (form 3) · proof
- tan 2x = 2 tan x / (1 − tan²x) — Double-angle identity for tangent · proof
When the angle is halved, or a square root of 1 ± cos appears
√((1 − cos x)/2) is sin(x/2) with a sign condition. If the angle range is not given, the ± is part of the answer, not a footnote.
- sin(x/2) = ±√((1 − cos x)/2) — Half-angle identity for sine · proof
- cos(x/2) = ±√((1 + cos x)/2) — Half-angle identity for cosine · proof
- tan(x/2) = (1 − cos x)/sin x — Half-angle identity for tangent (form 1) · proof
- tan(x/2) = sin x/(1 + cos x) — Half-angle identity for tangent (form 2) · proof
- tan(x/2) = csc x − cot x — Half-angle identity for tangent (form 3) · proof
When two different functions are multiplied
sin A cos B, cos A cos B, sin A sin B: a product of different functions rarely integrates, rarely solves, and always benefits from becoming a sum.
- sin A cos B = ½[sin(A + B) + sin(A − B)] — Product to sum: sin A cos B · proof
- cos A sin B = ½[sin(A + B) − sin(A − B)] — Product to sum: cos A sin B · proof
- cos A cos B = ½[cos(A − B) + cos(A + B)] — Product to sum: cos A cos B · proof
- sin A sin B = ½[cos(A − B) − cos(A + B)] — Product to sum: sin A sin B · proof
When the same function is added or subtracted
sin A + sin B, cos A − cos B: a sum of the same function becomes one product, which is what you need to factor an equation or evaluate a limit.
- sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2) — Sum to product: sin A + sin B · proof
- sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2) — Sum to product: sin A − sin B · proof
- cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2) — Sum to product: cos A + cos B · proof
- cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2) — Sum to product: cos A − cos B · proof
When a power of sine or cosine has to be integrated
Any power above one is reduced with the power-reducing identities: sin², cos², sin³, cos³, sin⁴, cos⁴ each become a sum of first powers of multiple angles.
- sin²x = (1 − cos 2x)/2 — Power-reducing formula for sin² · proof
- cos²x = (1 + cos 2x)/2 — Power-reducing formula for cos² · proof
- sin³x = (3 sin x − sin 3x)/4 — Reduction formula for sin³ · proof
- cos³x = (3 cos x + cos 3x)/4 — Reduction formula for cos³ · proof
- cos A cos B = ½[cos(A − B) + cos(A + B)] — Product to sum: cos A cos B · proof
- sin A sin B = ½[cos(A − B) − cos(A + B)] — Product to sum: sin A sin B · proof
When tan, sec, csc and cos are mixed together
Mixed function families are the usual reason a verification stalls. Convert everything to sin and cos first, then apply the ordinary rules — this is the standard advice for a reason.
- tan(x/2) = csc x − cot x — Half-angle identity for tangent (form 3) · proof
- tan A + tan B = sin(A + B) / (cos A cos B) — Tangent sum as a single fraction · proof
- tan θ = sin θ / cos θ — Quotient identity (tangent) · proof
- csc θ = 1/sin θ — Reciprocal identity (cosecant) · proof
- sec θ = 1/cos θ — Reciprocal identity (secant) · proof
- cot θ = 1/tan θ — Reciprocal identity (cotangent) · proof
- cot θ = cos θ / sin θ — Quotient identity (cotangent) · proof
When the angles add up to 90°
sin 32° and cos 58° are the same number. Complementary angles are what cofunction pairs describe.
- sin θ = cos(90° − θ) — Cofunction identity for sine · proof
- cos θ = sin(90° − θ) — Cofunction identity for cosine · proof
- tan θ = cot(90° − θ) — Cofunction identity for tangent · proof
When a negative angle appears
Which functions change sign under x → −x decides a lot of homework answers. Cosine and secant are even; sine, tangent, cotangent and cosecant are odd.
- sin(−θ) = −sin θ — Sine is odd · proof
- cos(−θ) = cos θ — Cosine is even · proof
- tan(−θ) = −tan θ — Tangent is odd · proof
When the letters are the angles and sides of one triangle
a, b, c and A, B, C are not independent: A + B + C = 180° and the sides follow the sine rule. Any identity in this family is only true under those constraints.
- ½ a b sin C = ½ b c sin A — Triangle area in two pairings · proof
What the tool does when it is unsure
Three outcomes, and the third one is deliberate:
- Something matches — you get the rule, the part of your expression that matched, and the expression after applying it.
- Something matched but did not help — press “I tried it” and the next candidate appears. Working through options is how the subject is actually done.
- Nothing matches — the tool says so and suggests the generic moves (write everything in sin and cos, clear a denominator). It will not invent an identity to look useful; a single made-up suggestion and the site stops being a reference.
Limits of this tool
- It matches structures, it does not simplify arbitrarily. Expressions needing algebra outside trig identities (expanding a cube, common denominators of three fractions) may show fewer options than a human would see.
- The catalogue holds 44 identities across 10 families — the ones listed above. Inverse-trig relations and hyperbolic identities are NOT in it yet, so the tool cannot suggest them; triangle relations are represented by the area identity at /identities/triangle.
- Angles are radians. Write
45°orpi/4, never a bare45.
Not sure the claim is even true? Verify two expressions — it shows a counterexample the moment one exists.