Common Trigonometric Identities You Should Know
Trig identities are equations that hold true for every angle, not just a lucky handful. They are the algebra of trigonometry: the legal moves that let you rewrite a tangled expression into something you can actually work with. You do not need to memorise hundreds of them — a small core set generates almost everything else, and nearly every identity you will ever meet in an introductory course can be derived, in a line or two, from the handful collected here.
The Pythagorean identities
The cornerstone comes straight from the unit circle, where a point (cos θ, sin θ) sits at distance 1 from the origin:
sin2θ + cos2θ = 1.
Divide every term by cos2θ and you get tan2θ + 1 = sec2θ. Divide instead by sin2θ and you get 1 + cot2θ = csc2θ. All three are really the same identity wearing different clothes.
The reciprocal identities
The three lesser-known functions are just upside-down versions of the familiar ones:
- csc θ = 1 / sin θ
- sec θ = 1 / cos θ
- cot θ = 1 / tan θ
A handy check for the confusable pair: cosecant goes with sine, and secant goes with cosine, even though the opening letters do not match.
The quotient identity
Tangent is not an independent function at all; it is a ratio of the other two:
tan θ = sin θ / cos θ,
and turning it over gives cot θ = cos θ / sin θ. Whenever a problem mixes tangent with sine and cosine, this identity lets you rewrite everything in one common language.
Even and odd
Flipping an angle to its negative reveals a symmetry. Cosine is even, so cos(−θ) = cos θ. Sine is odd, so sin(−θ) = −sin θ, and because tangent is their quotient it is odd too: tan(−θ) = −tan θ. These come directly from reflecting a point across the x-axis on the unit circle, which keeps its x-coordinate but negates its y-coordinate.
The cofunction identities
One more family falls straight out of the fact that a right triangle's two acute angles are complementary — they add up to 90°. Swap which acute angle you are standing at, and the sides labelled "opposite" and "adjacent" trade places, which turns sine into cosine and back again:
sin(90° − θ) = cos θ, and cos(90° − θ) = sin θ.
The same pairing holds for tangent and cotangent, and for secant and cosecant. It is why the two functions in each pair share a name with a "co-" prefix in the first place: cosine is literally the "complement's sine." A quick check confirms it: sin 30° = 0.5, and cos(90° − 30°) = cos 60° = 0.5 as well. The same relationship is exactly why the exact-value table for 30° and 60° look like mirror images of each other, with sine and cosine simply swapping places between the two angles.
The angle-sum formulas
Two more identities let you break a sum of angles apart:
sin(A + B) = sin A cos B + cos A sin B,
cos(A + B) = cos A cos B − sin A sin B.
Watch the cosine version carefully: the sign in the middle flips to a minus, which trips up a lot of students. The matching difference formulas simply reverse the inner signs: sin(A − B) = sin A cos B − cos A sin B, and cos(A − B) = cos A cos B + sin A sin B. Set A equal to B in the sum versions and you fall straight into the double-angle results, such as sin 2A = 2 sin A cos A, so these two lines quietly generate a whole family of further identities — most of what looks like a separate formula to memorise is really just one of these two equations wearing a disguise.
An identity holds for every angle; an equation doesn't
It is worth being precise about what makes something an identity rather than an ordinary equation. sin θ = 0.5 is a perfectly good equation, but it is only true for specific angles (30°, 150°, and their coterminal repeats) — solving it is the whole point of inverse trig. sin2θ + cos2θ = 1, on the other hand, is true for literally every real value of θ, with no solving required and no exceptions to check for. That universality is exactly what makes identities safe to substitute into a larger expression at any point: you are never narrowing down which angles are allowed, only rewriting the same quantity in a different, equally valid form.
What identities are actually for
Beyond memorising them, it helps to know why they earn their place. Their day job is simplification: replacing a stubborn expression with an equivalent one that a problem can actually use. A term like 1 − sin2θ is far friendlier written as cos2θ, and an integral or equation bristling with tangents often melts away once you rewrite everything as sines and cosines. In calculus, physics, and engineering, most trig identities are used not to prove abstract equalities but to clear a path to an answer.
A bonus: the half-angle formula
The double-angle identity can be turned around to reach the opposite direction: instead of doubling an angle, it can halve one. Starting from the cosine double-angle identity in its cos2 form, cos 2A = 2cos2A − 1, rearranging for cos A gives cos A = ±√((1 + cos 2A) / 2). Relabelling 2A as θ (so A = θ/2) turns this into a formula for the cosine of a half-angle purely in terms of the full angle's own cosine: cos(θ/2) = ±√((1 + cos θ) / 2), with the sign chosen to match whichever quadrant θ/2 actually falls in. Checking with θ = 30°, so θ/2 = 15°: cos 30° ≈ 0.866, and √((1 + 0.866) / 2) = √0.933 ≈ 0.9659, matching cos 15° ≈ 0.9659 exactly. The sine half-angle formula follows the identical pattern from the other cos 2A identity (cos 2A = 1 − 2sin2A): sin(θ/2) = ±√((1 − cos θ) / 2).
Worked verification: sin 75° exactly
The angle 75° is not on the standard table, but it splits neatly as 45° + 30°, both of which we know exactly. Apply the sine sum formula:
sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°.
Substitute the exact values: (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4.
Now turn it into a decimal to confirm: √6 ≈ 2.449 and √2 ≈ 1.414, so (2.449 + 1.414)/4 ≈ 3.863/4 ≈ 0.9659. A calculator gives sin 75° ≈ 0.9659 as well, so the identity checks out perfectly. As a second quick sanity test, the Pythagorean identity at 30° gives (1/2)2 + (√3/2)2 = 1/4 + 3/4 = 1, exactly as promised.
A quick-reference table
Collected in one place, the identities above cover the vast majority of what a first pass through trigonometry actually needs:
| Family | Identity |
|---|---|
| Pythagorean | sin2θ + cos2θ = 1 |
| Pythagorean | tan2θ + 1 = sec2θ |
| Pythagorean | 1 + cot2θ = csc2θ |
| Reciprocal | csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ |
| Quotient | tan θ = sin θ / cos θ |
| Cofunction | sin(90° − θ) = cos θ, cos(90° − θ) = sin θ |
| Even/odd | cos(−θ) = cos θ, sin(−θ) = −sin θ |
| Angle sum | sin(A ± B) = sin A cos B ± cos A sin B |
| Angle sum | cos(A ± B) = cos A cos B ∓ sin A sin B |
| Double angle | sin 2A = 2 sin A cos A |
A one-line taste of simplification
Before the dedicated walkthrough linked below, here is the shortest possible example of an identity earning its keep. The expression sin θ × cot θ looks like it needs two separate pieces of information, but substitute the quotient identity cot θ = cos θ / sin θ and the sin θ terms cancel outright: sin θ × (cos θ / sin θ) = cos θ. What began as a product of two different functions collapses to a single cosine. Checking at θ = 35°: sin 35° ≈ 0.5736 and cot 35° ≈ 1.4281, and their product is 0.5736 × 1.4281 ≈ 0.8192 — matching cos 35° ≈ 0.8192 exactly.
The double-angle formula, checked numerically
Setting A = B inside the sine sum formula gives the double-angle identity sin 2A = 2 sin A cos A directly: sin(A + A) = sin A cos A + cos A sin A = 2 sin A cos A. It is worth confirming with real numbers rather than trusting the algebra alone. At A = 25°, evaluating the six functions gives sin 25° ≈ 0.4226 and cos 25° ≈ 0.9063, so 2 × 0.4226 × 0.9063 ≈ 0.766. Evaluating sine directly at the doubled angle, 50°, gives sin 50° ≈ 0.766 as well — the two routes agree. This is exactly the kind of numeric double-check worth running on any identity before trusting it inside a longer calculation, since it catches sign errors and misremembered formulas immediately.
From "know it" to "use it"
Listing the identities is only half the job; the more valuable skill is reaching for the right one to simplify a specific expression, which is a different kind of practice from memorising a table. Using Trig Identities to Actually Simplify an Expression walks through that process step by step on a handful of concrete expressions, including a numeric check at every step so you can see exactly why each substitution is legal. The trig function & identity calculator is also useful while practising: it evaluates all six functions at once and reports the Pythagorean identity's live value, so you can instantly check whether a simplification you have done by hand actually holds at a specific angle, at any angle you choose, not just the handful worked through in this article or the more detailed walkthrough it links to.
Identities become second nature once you use them. Put them to work with the calculators on SinCosTan.