Every trigonometry student eventually faces the demand to prove that one expression equals another, and the identities are the allowed moves. The most famous, the Pythagorean identity, says sin squared plus cos squared equals 1 for every angle: at 30 degrees, 0.5 squared is 0.25 and 0.8660 squared is 0.75, and 0.25 plus 0.75 equals 1, exactly. Related identities rewrite tangent as sine over cosine, express doubled angles in terms of single angles, and flip between sums and products, the toolkit behind simplifying integrals, analyzing alternating current circuits, and tuning musical instruments. An identity is an equation that holds for every angle where both sides are defined, which is what separates it from an ordinary equation that holds only for particular angles, and that universality is what makes identities safe to use as rewriting rules inside a proof.

The Pythagorean family, and where it comes from

The Pythagorean identity comes straight from the unit circle, the circle of radius 1 where every angle lands on the point (cos, sin). By the Pythagorean theorem, the squares of those coordinates add to the square of the radius, which is 1, so sin squared plus cos squared equals 1. Check it at three anchor angles and one messy angle, because a rule that only works at anchors is not a rule. At 30 degrees: 0.25 plus 0.75 equals 1. At 45 degrees: sine and cosine are both 0.7071, each squared is 0.5, and 0.5 plus 0.5 equals 1. At 60 degrees: sine squared is 0.75 and cosine squared is 0.25, and the sum is 1 again, the 30-degree pair with the roles swapped. At 37 degrees, an angle with no tidy values, sine is about 0.6018 and cosine is about 0.7986, their squares are about 0.3622 and 0.6378, and the sum is 1.0000 to rounding, which is how you know the identity is a theorem and not a coincidence of nice angles. Dividing the Pythagorean identity by cos squared gives the second member of the family, 1 plus tan squared equals sec squared: at 30 degrees, tan squared is about 0.3333 and sec squared is about 1.3333, and 1 plus 0.3333 equals 1.3333. Dividing by sin squared gives the third, 1 plus cot squared equals csc squared: at 60 degrees, cot squared is about 0.3333 and csc squared is about 1.3333, and again the two sides agree. Memorizing all three is less work than it looks, because the second and third are the first identity divided through, not three separate facts.

Reciprocal, quotient, co-function, and even-odd identities

The reciprocal identities say each function is one over its partner: cosecant is 1 over sine, secant is 1 over cosine, cotangent is 1 over tangent, and each statement reads equally well in reverse. The quotient identities say tangent is sine over cosine and cotangent is cosine over sine, and they are the workhorses of proofs, because they convert every tangent and cotangent into sines and cosines, the two functions the Pythagorean identity actually constrains. Check the quotient at 30 degrees: sine over cosine is 0.5 divided by 0.8660, which is about 0.5774, exactly tan of 30 degrees, and cosine over sine is 0.8660 divided by 0.5, which is 1.7321, exactly cot of 30 degrees. The co-function identities say sine of an angle equals cosine of its complement: sin of 30 degrees equals cos of 60 degrees, both 0.5, because the two acute angles of a right triangle trade the roles of opposite and adjacent. The even-odd identities describe reflection: cosine is even, so cos of negative 30 degrees equals cos of 30 degrees, 0.8660, while sine and tangent are odd, so sin of negative 30 degrees is negative 0.5. Together these four families let a proof move any trig expression toward sines and cosines of a single angle, which is the standard opening move of nearly every identity proof.

Sum and difference formulas: exact values for angles you cannot look up

The sum and difference formulas find trig values of combined angles from the values of the parts, and they are how exact values for 75 degrees and 15 degrees are built from the anchor angles 45 and 30. Sine of a sum is sine of the first times cosine of the second plus cosine of the first times sine of the second. For 75 degrees as 45 plus 30: 0.7071 times 0.8660 is about 0.6124, and 0.7071 times 0.5 is about 0.3536, and their sum is about 0.9659, which is sin of 75 degrees. Cosine of a difference is cosine of the first times cosine of the second plus sine of the first times sine of the second, note the sign flip: cos of 75 degrees as cos of 45 minus 30 is 0.7071 times 0.8660 minus 0.7071 times 0.5, which is 0.6124 minus 0.3536, about 0.2588. Sine of 15 degrees as sin of 45 minus 30 is the same two products subtracted, 0.6124 minus 0.3536, about 0.2588, which is also cos of 75 degrees, as the co-function identities require, a satisfying cross-check. The tangent sum formula carries a denominator: tan of a sum is the sum of the tangents divided by 1 minus their product, and the minus sign in that denominator is the detail most often dropped. These formulas are also the reason the double-angle formulas exist at all: set the two angles equal and the sum formulas collapse into them.

Double-angle, half-angle, and power-reducing formulas

The double-angle formulas turn the sine, cosine, or tangent of a doubled angle into expressions in the single angle, and they are verified most convincingly by running them at 30 degrees doubled to 60. Sine of double the angle is 2 times sine times cosine: 2 times 0.5 times 0.8660 is 0.8660, which is sin of 60 degrees. Cosine of double the angle has three equivalent forms, and choosing among them is a proof skill in itself: cos squared minus sin squared is 0.75 minus 0.25, which is 0.5; 2 times cos squared minus 1 is 2 times 0.75 minus 1, which is 0.5; and 1 minus 2 times sin squared is 1 minus 2 times 0.25, which is 0.5, all three equal to cos of 60 degrees. Tangent of double the angle is 2 times tan divided by 1 minus tan squared: 2 times 0.5774 divided by 1 minus 0.3333 is 1.1547 divided by 0.6667, about 1.7321, which is tan of 60 degrees. The half-angle formulas run the other direction and carry a sign warning: sine of half an angle is plus or minus the square root of 1 minus cos over 2, and the sign depends on which quadrant the half angle lands in, not on the original angle. Sine of 15 degrees as half of 30 is the square root of 1 minus 0.8660 over 2, the square root of 0.0670, about 0.2588, positive because 15 degrees sits in the first quadrant. Cosine of 15 degrees is the square root of 1 plus 0.8660 over 2, the square root of 0.9330, about 0.9659, matching the sum-formula value from earlier. The power-reducing formulas are the half-angle formulas rearranged: sin squared of 30 degrees equals 1 minus cos of 60 degrees over 2, which is 1 minus 0.5 over 2, which is 0.25, and cos squared of 30 degrees equals 1 plus cos of 60 degrees over 2, which is 0.75. Calculus students meet power-reducing constantly, because it turns a squared trig function, which is hard to integrate, into a first-power cosine, which is easy.

A full proof, worked slowly

Here is a complete proof of the identity tan plus cot equals sec times csc, done the standard way: convert everything to sines and cosines, combine over a common denominator, and let the Pythagorean identity finish the job. Start with the left side, tan plus cot, and rewrite it as sine over cosine plus cosine over sine. The common denominator is sine times cosine, so the sum becomes sine squared plus cosine squared, all over sine times cosine. The numerator is the Pythagorean identity, exactly 1, so the left side simplifies to 1 over sine times cosine. The right side, sec times csc, is 1 over cosine times 1 over sine, which is the same 1 over sine times cosine, so the two sides are equal and the identity is proved. Check it numerically at 30 degrees to make sure the algebra did not smuggle in an error: tan of 30 is 0.5774 and cot of 30 is 1.7321, and their sum is 2.3094, while sec of 30 is 1.1547 and csc of 30 is 2, and their product is 2.3094, agreeing to rounding. A second, shorter simplification shows the same method at smaller scale: sine times secant is sine times 1 over cosine, which is sine over cosine, which is tangent, and at 30 degrees that is 0.5 times 1.1547, about 0.5774, tan of 30 degrees. The method generalizes: when a proof stalls, convert to sines and cosines, find a common denominator, and look for a Pythagorean numerator.

How to use this calculator, honestly

This calculator handles the squaring step at the heart of verifying the Pythagorean identity, term by term. Type sin of your angle in both the Variable A box and the Variable B box and the Result shows sin squared: 0.5 twice gives 0.25 for 30 degrees. Repeat for cos, 0.8660 twice gives 0.75, and add the two Results yourself: 0.25 plus 0.75 equals 1, confirming the identity. At 45 degrees, 0.7071 twice each way gives 0.5 and 0.5, again summing to 1. At 60 degrees, the squares swap to 0.75 and 0.25 and still sum to 1. The tool squares one term at a time; the addition, the quotient steps, and the angle-doubling arithmetic stay in your hands, which is the point, because a proof is a chain of steps you can defend, not a single button press. Before any of it, check your calculator's mode: school problems use degrees, so sin of 30 degrees must be 0.5, while a calculator left in radian mode computes sin of 30 radians, about negative 0.988, and every identity check built on that value will fail for a reason that has nothing to do with the identity.

Five mistakes that cost real marks

The first mistake is dividing by zero in disguise: tan of 90 degrees and sec of 90 degrees are undefined because cos of 90 degrees is 0, and an identity proved by dividing through by cos silently assumes cos is not zero. The second is taking the wrong sign in a half-angle formula, because the sign follows the quadrant of the half angle, not the sign of the cosine inside the square root. The third is mixing degrees and radians, the silent killer: 45 degrees is 0.7854 radians, and a sine value computed in the wrong mode poisons every step after it. The fourth is squaring both sides of an equation during a proof and forgetting that squaring can introduce extraneous solutions, which must be checked against the original equation. The fifth is memorizing the double-angle formulas as three unrelated facts instead of deriving them from the sum formulas, which triples the memory load and guarantees that one of the three will be misremembered under exam pressure. Each of these is caught by the same two habits: verify numerically at 30, 45, and 60 degrees, and never divide by a trig function without asking where it could be zero.

Where identities actually get used

Identities are not homework decoration; they are the rewriting rules of every field that uses waves. In alternating current circuits, voltages and currents are sines and cosines with phase shifts, and the sum formulas combine them into a single wave whose amplitude and phase can be read off. In calculus, the power-reducing formulas turn sin squared and cos squared into forms that integrate directly, and the double-angle formulas differentiate cleanly where the original expression does not. In signal processing and acoustics, product-to-sum formulas turn a product of two waves, which is hard to analyze, into a sum of two waves at sum and difference frequencies, which is how beat frequencies and sidebands are understood. In computer graphics, rotation matrices are built from sines and cosines, and the sum formulas are what make composing two rotations equivalent to a single rotation by the sum of the angles. A student who can prove tan plus cot equals sec times csc has the same skill a circuit engineer uses daily: rewrite until the expression says something measurable.

Related calculators on this site

If trigonometric identities are part of a bigger job, three companion tools carry the next step. The trigonometry calculator works the right-triangle ratios, sine, cosine, and tangent from sides and angles, the raw values every identity check consumes. The triangle angles calculator handles the angle-sum and law-of-sines routes when the triangle itself, rather than an identity, is the unknown. The unit circle calculator places every angle on the point (cos, sin) from which the Pythagorean identity falls out directly, and the sine-wave and angle tools on this site cover the wave applications where sum and product formulas earn their keep. Each one uses the same discipline: check the degree mode first, verify at 30, 45, and 60 degrees, and never divide by a trig function without checking where it is zero.

A classroom note on proving versus checking

Students sometimes ask whether checking an identity at 30, 45, and 60 degrees counts as proving it. It does not, and the distinction matters: a check at three angles shows the identity is plausible and catches arithmetic slips, while a proof shows it holds for every angle, which is what the word identity promises. The proof above works because it uses only rewriting rules that are themselves true for every angle, the quotient definitions and the Pythagorean identity, and ends with both sides visibly identical. That is the standard to aim for: check numerically to find errors, then prove symbolically to claim the result, and keep the two activities in that order, because a proof written before the numbers are checked usually hides the slip instead of exposing it.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Trigonometric Identities Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.