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A2 Level · Pure Mathematics 3 WMA13

Trigonometric Proof

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What You'll Learn

Proving trigonometric identities isn't about magical breakthroughs—it's about having a clear strategy and the toolkit to execute it. This chapter teaches you how to:

Recognise which trigonometric identities to use and when
Manipulate one side of an identity step-by-step until it matches the other
Use compound angle formulae as your secret weapon for harder proofs
Handle complex fractions with confidence
Know whether to start from the left or right side for efficiency

Big Idea: A trigonometric proof is like following a recipe with substitutions. You start with one form, apply known identities (like ingredients), simplify (blend the mixture), and arrive at your target. The key is knowing which identity to use next.

The Fundamental Approach to Proving Identities

Step-by-Step Strategy

When you're asked to prove that one expression equals another, follow this method:

Choose your starting side. Usually the left-hand side (LHS), but sometimes the right-hand side (RHS) is simpler.
Identify your target. Keep the other side visible—it guides which identities will help you.
Pick an identity to apply. Choose one that simplifies your expression or moves it closer to the target.
Apply it carefully. Substitute or manipulate, one step at a time. Show every algebraic step.
Repeat until you reach the target. When your expression matches the other side, you're done.

Key principle: You never write an equals sign between the two sides until you've proved they're equal. Write the side you're working on, then manipulate it step by step until it becomes the target expression.

The Identities You Must Know

These are your toolkit. Commit them to memory (or know where to find them in the formula booklet):

sin²θ + cos²θ = 1

sin2θ = 2sinθ cosθ
cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
tan2θ = 2tanθ / (1 − tan²θ)

tanθ = sinθ / cosθ
cotθ = cosθ / sinθ
secθ = 1 / cosθ
cosecθ = 1 / sinθ

cos(A − B) = cosA cosB + sinA sinB
sin(A − B) = sinA cosB − cosA sinB

Exam note: The formula booklet gives you many identities, but not all. You're expected to know the ones listed above, plus be able to derive others from them.

Worked Example 1: Using Double Angle & Pythagorean Identity

Show that tanθ + cotθ = 2cosec2θ

Strategy: The right side is cosec2θ (which involves sin2θ), so we'll probably need the double angle formula sin2θ = 2sinθ cosθ. Start from the left side (it has tan and cot, which are easier to manipulate).

Step 1: Write tan and cot in terms of sin and cos

tanθ + cotθ = sinθ/cosθ + cosθ/sinθ

We always convert to sin and cos first—this is your foundation for everything.

Step 2: Combine over a common denominator

= sin²θ/sinθ cosθ + cos²θ/sinθ cosθ = (sin²θ + cos²θ) / (sinθ cosθ)

Step 3: Apply the Pythagorean identity sin²θ + cos²θ = 1

= 1 / (sinθ cosθ)

Step 4: Use the double angle formula sin2θ = 2sinθ cosθ

= 1 / (½ sin2θ) = 2 / sin2θ = 2cosec2θ ✓

Done! We transformed the left side into the right side.

What Just Happened?

Notice the flow:

We converted tan and cot to sin and cos (a universal first move in trig proofs).
We found a common denominator—this revealed the Pythagorean identity.

Lesson:

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Also in the full note

Worked Example 2: Compound Angle Formulae & Strategic Choices
Handling Complex Fractions
Practice Questions
Common Mistakes & How to Avoid Them
What to Memorise
Concepts Checklist
Exam Tips & Tricks
Example: The Fraction-Within-Fraction Technique

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Trigonometric Proof