Trig Identity Calculator

How to Use This Calculator

1. Enter an angle and select an identity type to verify it numerically.
2. Use the Sum/Difference tab to enter two angles A and B.
3. Use Double Angle for sin(2A), cos(2A), and tan(2A).
4. Use Professional for product-to-sum, sum-to-product, and Mollweide formulas.

Formula

Example

Frequently Asked Questions

• What are the Pythagorean identities? ▼
  The three Pythagorean identities are: (1) sin²(θ) + cos²(θ) = 1, (2) 1 + tan²(θ) = sec²(θ), and (3) 1 + cot²(θ) = csc²(θ). All three derive from the unit circle, where a point on the circle satisfies x² + y² = 1 with x = cos(θ) and y = sin(θ). Dividing identity (1) by cos²(θ) gives identity (2); dividing by sin²(θ) gives identity (3). Example: at θ = 30°, sin(30°) = 0.5 and cos(30°) ≈ 0.866, so (0.5)² + (0.866)² = 0.25 + 0.75 = 1. A common pitfall is forgetting to square each term — sin(θ) + cos(θ) does NOT equal 1; only the sum of their squares does.
• What is the double angle formula for sin? ▼
  The double angle formula for sine is sin(2θ) = 2·sin(θ)·cos(θ). It is derived from the sum formula sin(A + B) by setting A = B = θ. For cosine there are three equivalent forms: cos(2θ) = cos²(θ) − sin²(θ) = 2cos²(θ) − 1 = 1 − 2sin²(θ). Example: θ = 45°, sin(2 × 45°) = sin(90°) = 1; checking: 2·sin(45°)·cos(45°) = 2·(√2/2)·(√2/2) = 2·(1/2) = 1. A common mistake is writing sin(2θ) = 2·sin(θ) — that drops the cosine factor and gives wrong results for all angles except 0°. The double-angle identities are essential for simplifying integrals like ∫cos²(x) dx.
• What is the sum formula for sin(A+B)? ▼
  The angle addition formula for sine is sin(A + B) = sin(A)cos(B) + cos(A)sin(B). The corresponding formula for cosine is cos(A + B) = cos(A)cos(B) − sin(A)sin(B). Note the sign difference: cosine subtraction versus sine addition. Example: sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°) = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 ≈ 0.9659. A common error is using sin(A + B) = sin(A) + sin(B), which is wrong — for example sin(90°) ≠ sin(45°) + sin(45°). These formulas are foundational for deriving all other compound-angle results.
• What are product-to-sum formulas? ▼
  Product-to-sum formulas convert products of trig functions into sums, which is useful for integration. The key formulas are: sin(A)cos(B) = ½[sin(A + B) + sin(A − B)]; cos(A)cos(B) = ½[cos(A − B) + cos(A + B)]; sin(A)sin(B) = ½[cos(A − B) − cos(A + B)]. Example: sin(75°)cos(15°) = ½[sin(90°) + sin(60°)] = ½[1 + 0.866] ≈ 0.933. These identities are essential in calculus when integrating products such as ∫sin(3x)cos(5x) dx, which can be rewritten as ½∫[sin(8x) + sin(−2x)] dx. A common pitfall is mixing up the signs in the sin·sin formula — it uses a minus sign after the cosine terms.
• How are trig identities used in calculus? ▼
  Trigonometric identities appear throughout calculus as simplification tools. In integration, the Pythagorean identity rewrites ∫sin²(x) dx using cos(2x) = 1 − 2sin²(x), giving sin²(x) = (1 − cos(2x))/2, so ∫sin²(x) dx = x/2 − sin(2x)/4 + C. Product-to-sum formulas allow integrating products like ∫sin(3x)cos(x) dx directly. In differential equations, trig identities simplify particular solutions. In Fourier analysis, identities confirm orthogonality of sin and cos functions over [0, 2π]. Example pitfall: students sometimes apply ∫sin²(x) dx = −cos²(x)/2 + C — that is incorrect because d/dx[−cos²(x)/2] = cos(x)sin(x), not sin²(x). Always apply the double-angle substitution first.

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