Proving Trig Identities

Key idea

Transform one side of an equation into the other.

In depth

To prove an identity, work on one side (usually the more complicated) until it matches the other, without moving terms across. Strategies: rewrite in sin and cos, use Pythagorean identities, combine fractions.

Worked example

(1 − cos²x)/sin x = sin²x/sin x = sin x.

Real-world uses

• Checking formulas before using them in engineering calculations and exam proofs

Common mistake

Treating it as an equation and manipulating both sides together.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Trigonometry

• Graph of Tangent – Periodic with vertical asymptotes at odd multiples of π/2.
• Phase Shift – Horizontal translation of a wave.
• Vertical Shift and Midline – Moves the wave up or down.
• General Sinusoid – Combine amplitude, period, phase and shift.
• Graphs of Reciprocal Trig Functions – Secant, cosecant and cotangent graphs.
• Pythagorean Identity – The key identity from the unit circle.