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.

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 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.

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.

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.

Identities become second nature once you use them. Put them to work with the calculators on SinCosTan.