The Identities Worth Knowing
| Family | Identity |
|---|---|
| Pythagorean | sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = csc²x |
| Quotient | tan x = sin x ÷ cos x, cot x = cos x ÷ sin x |
| Reciprocal | csc x = 1/sin x, sec x = 1/cos x, cot x = 1/tan x |
| Double angle | sin 2x = 2 sin x cos x, cos 2x = cos²x − sin²x = 2cos²x − 1 = 1 − 2sin²x |
| Sum | sin(A + B) = sin A cos B + cos A sin B, cos(A + B) = cos A cos B − sin A sin B |
The Pythagorean identity is just x² + y² = 1 on the unit circle, where x = cos θ and y = sin θ. Divide it by cos²x and you get 1 + tan²x = sec²x; divide by sin²x and you get the cotangent version. You only ever need to memorise one.
How to Prove a Trig Identity
Pick the more complicated side and transform it until it matches the other. Do not move terms across the equals sign, because that assumes the thing you are trying to show. Rewriting everything in sine and cosine is the safest opening move.
tan x + cot x = sec x csc x
- Rewrite the left side in sine and cosine: sin x/cos x + cos x/sin x.
- Use the common denominator sin x cos x: (sin²x + cos²x)/(sin x cos x).
- Apply the Pythagorean identity: the numerator is 1.
- That gives 1/(sin x cos x) = (1/cos x)(1/sin x).
- Which is sec x · csc x, the right side. ∎
Answer: Proved
Double-Angle Formulas and Where They Come From
Set B = A in the sum formulas and the double-angle formulas drop out: sin 2A = sin A cos A + cos A sin A = 2 sin A cos A. The three forms of cos 2A come from swapping cos² or sin² using the Pythagorean identity, so you can always pick the one that suits the problem.
sin(2x)/(2 sin x)
- Replace sin 2x with 2 sin x cos x.
- The expression becomes (2 sin x cos x) ÷ (2 sin x).
- Cancel the common factor 2 sin x, which is valid when sin x ≠ 0.
Answer: cos x
Sum and Difference Formulas: Exact Values for 75°
sin 75°
- Split the angle into ones with known values: 75° = 45° + 30°.
- Use sin(A + B) = sin A cos B + cos A sin B.
- Substitute: sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2).
- Multiply: √6/4 + √2/4.
Answer: (√6 + √2)/4 ≈ 0.9659
The Identities That Do Most of the Work
| Family | Identity |
|---|---|
| Pythagorean | sin²x + cos²x = 1; 1 + tan²x = sec²x; 1 + cot²x = csc²x |
| Double angle | sin 2x = 2 sin x cos x; cos 2x = cos²x − sin²x = 1 − 2 sin²x |
| Sum | sin(a + b) = sin a cos b + cos a sin b |
| Quotient | tan x = sin x ÷ cos x |
Quick numeric check: at x = π/6, sin x = 0.5, so sin 2x should equal 2 × 0.5 × 0.866 = 0.866, and it does (sin 60° = 0.866).
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: memorising dozens of identities. Four families cover most proofs.
Underrated: rewriting everything in sine and cosine. It is the single most reliable first move.
Plain wrong: working on both sides at once. In a proof you start from one side and transform it into the other. Doing both sides assumes what you want to show.
Where Identities Get Used, and Mistakes
In calculus, integrating sin²x needs the half-angle form (1 − cos 2x)/2. In physics, adding two waves of the same frequency uses the sum formulas. Signal processing leans on them every day.
- Treating sin(A + B) as sin A + sin B. It is not.
- Writing sin²x as sin(x²). The square belongs to the value, not the angle.
- Working on both sides of the equals sign at once.
- Cancelling a factor that might be zero without noting the restriction.
Why Students Get Stuck on Trigonometric Identities, and What a Good Calculator Should Do
Searching for help with trigonometric identities usually starts late at night with one problem that will not work out. Proving an identity needs a plan, not a formula. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to turn one side into the other with known identities, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Not knowing which side to start withSimplify sin(x)^2 + cos(x)^2↗
- Mixing up the Pythagorean identitiesSimplify sin(2x)/sin(x)↗
- Working on both sides at onceSimplify (1 - cos(x)^2)/sin(x)↗
- Dropping a sign with double anglesSimplify tan(x) * cos(x)↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants the identity used at each step named
- Wants the proof checked numerically
- Wants simplification as well as verification
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Simplify sin(x)^2 + cos(x)^2↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.