Mathematics
Trigonometric Identities
Trigonometric identities are equations that hold for every angle. Knowing the core set (Pythagorean, reciprocal, quotient, double-angle, sum/difference) lets you simplify any trig expression or solve any trig equation.
How to Approach Trigonometric Identities
- 01
Start with what's given
Look at the most complicated side. Convert everything to sin and cos using reciprocal and quotient identities.
- 02
Apply the right identity
Pythagorean: sin² + cos² = 1. Double-angle: sin(2x) = 2sin(x)cos(x). Sum: sin(a±b) = sin(a)cos(b) ± cos(a)sin(b). Pick the one that simplifies your expression.
- 03
Verify both sides match
Keep simplifying until both sides of the identity are clearly equal. Don't 'move terms across' — that's algebra. Identities prove by transformation, one side at a time.
Frequently Asked Questions
Do I need to memorize every identity?
Memorize the Pythagorean (sin²+cos²=1), the reciprocals (csc=1/sin, sec=1/cos, cot=1/tan), and the sum/difference formulas. Double-angle and half-angle derive from those.
What's the trick to proving an identity?
Work on the more complicated side. Convert everything to sin and cos first. Look for Pythagorean patterns that let you substitute sin² = 1 - cos².
When are trig identities used in real problems?
Calculus integration, physics waves, electrical engineering, computer graphics rotations. Most simplifications in calculus require a trig identity at some step.
Related Topics
More step-by-step guides in Mathematics and adjacent subjects.
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