Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that remain true for all values of the variable where both sides are defined. The most fundamental identity is sin²x + cos²x = 1, known as the Pythagorean identity. These relationships form the foundation for simplifying complex trigonometric expressions and solving equations across mathematics, physics, and engineering.
Why it matters
Trigonometric identities appear throughout advanced mathematics and real-world applications. In electrical engineering, power calculations use the identity cos²θ + sin²θ = 1 to analyze alternating current circuits, where θ represents phase angles. Signal processing relies on identities to simplify expressions in Fourier transforms, which decompose complex waveforms into component frequencies. In physics, wave interference patterns require simplification using identities like sin(A ± B) = sin A cos B ± cos A sin B. Calculus integration often depends on identities to transform integrals into solvable forms, such as converting ∫sin²x dx using the identity sin²x = (1 - cos 2x)/2. Architecture and construction use these relationships in structural analysis, where forces are resolved into components using trigonometric ratios and their identities.
How to solve trigonometric identities
Trig Identities — Simplify
- 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.
- Rewrite in terms of sin and cos, then cancel or apply Pythagorean.
Example: (1 − sin²x)·sec x = cos²x · (1/cos x) = cos x.
Worked examples
Verify the Pythagorean identity sin²θ + cos²θ = 1 at θ = 60°. Show that sin²(60°) + cos²(60°) equals 1.
Answer: 1
- Recall the exact values of sin(60°) and cos(60°) → sin(60°) = √32, cos(60°) = 12 — These are the standard values you memorise from the unit circle.
- Square each value → sin²(60°) = 34, cos²(60°) = 14 — Squaring a fraction squares both numerator and denominator.
- Add the two squared values → 34 + 14 = 1 — The sum always equals 1 for any angle θ — this is the Pythagorean identity, and it comes from the fact that any point (cos θ, sin θ) on the unit circle satisfies x² + y² = 1.
Simplify the expression: 1/sin x
Answer: csc x
- Identify which identity applies → Use: Reciprocal identity — Look for the shape of the expression. Pythagorean, quotient, and reciprocal identities each have a recognisable form.
- Apply the identity → 1/sin x = csc x — Rewriting using the reciprocal identity gives the simplified form.
Simplify the expression: (1 - cos²x)/sin x
Answer: sin x
- Rewrite using basic identities → 1 − cos²x = sin²x, then sin²x/sin x = sin x — Combine the quotient, reciprocal, and Pythagorean identities until the expression reduces to a single trig function or a constant.
- State the simplified result → (1 - cos²x)/sin x = sin x — Verify by substituting a specific value of x (e.g. π/4) on both sides.
Common mistakes
- Confusing the Pythagorean identity by writing sin²x + cos²x = 0 instead of sin²x + cos²x = 1, leading to incorrect simplifications.
- Incorrectly applying reciprocal identities, such as writing sec x = sin x instead of sec x = 1/cos x, which produces wrong values.
- Mixing up quotient identities by writing tan x = cos x/sin x instead of tan x = sin x/cos x, resulting in the reciprocal of the correct answer.