Trigonometric Identities
Trigonometric identities are fundamental equations that express relationships between sine, cosine, tangent, and other trigonometric functions. These identities hold true for all permissible values of the variable and serve as powerful tools for simplifying expressions and solving equations. The most famous is the Pythagorean identity: sin²θ + cos²θ = 1, which stems directly from the unit circle definition of trigonometric functions.
Why it matters
Trigonometric identities appear throughout A-level mathematics and beyond, forming the foundation for calculus integration techniques and differential equations. Engineers use these identities when analysing alternating current circuits, where voltage and current follow sinusoidal patterns — for instance, calculating power consumption requires identities like cos²θ = (1 + cos(2θ))/2. In signal processing, identities help decompose complex waveforms into simpler components. Computer graphics relies on angle addition formulae to rotate 3D objects efficiently. Physics applications include wave interference patterns and harmonic motion analysis. Students encounter these identities in Year 12 for basic verification and simplification, then progress to proving complex identities using double angle formulae in Year 13. Mastering these relationships builds essential algebraic manipulation skills needed for higher mathematics courses.
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/tan x
Answer: cot 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/tan x = cot x — Rewriting using the reciprocal identity gives the simplified form.
Simplify the expression: sin x · cot x
Answer: cos x
- Rewrite using basic identities → cot = cos/sin, so sin · cot = sin · (cos/sin) = cos — Combine the quotient, reciprocal, and Pythagorean identities until the expression reduces to a single trig function or a constant.
- State the simplified result → sin x · cot x = cos x — Verify by substituting a specific value of x (e.g. π/4) on both sides.
Common mistakes
- Confusing the reciprocal relationships leads to errors like writing csc x = sin x instead of csc x = 1/sin x, or incorrectly stating that sec(30°) = 1/2 when it actually equals 2/√3.
- Incorrectly applying the Pythagorean identity by writing sin²x - cos²x = 1 instead of sin²x + cos²x = 1, which produces wrong results like claiming sin²(45°) - cos²(45°) = 1 when it actually equals 0.
- Mishandling quotient identities by writing tan x = cos x/sin x instead of tan x = sin x/cos x, leading to errors like calculating tan(60°) = (1/2)/(√3/2) = 1/√3 instead of the correct value √3.