Why Identity Proofs Matter#
Proving identities is one of the most frequently examined topics in Paper 2. You will see at least one “Prove that…” question worth 5–8 marks. The good news: there is a clear strategy that works every time.
1. The Golden Rule#
Work with ONE side only. Choose the more complicated side and manipulate it until it looks like the other side.
You are NOT solving an equation. You are showing that the Left Hand Side (LHS) and Right Hand Side (RHS) are always equal, for every value of $\theta$.
Never cross the equals sign. Never move terms from one side to the other.
2. The Toolkit#
Every identity proof uses some combination of these tools:
Tool 1: The Quotient Identity#
$$ \tan\theta = \frac{\sin\theta}{\cos\theta} $$When to use it: Whenever you see $\tan$ in a proof, replace it immediately. Working with $\sin$ and $\cos$ only makes everything simpler.
Tool 2: The Pythagorean Identity#
$$ \sin^2\theta + \cos^2\theta = 1 $$This can be rearranged:
- $\sin^2\theta = 1 - \cos^2\theta$
- $\cos^2\theta = 1 - \sin^2\theta$
When to use it: When you see $1 - \sin^2\theta$ or $1 - \cos^2\theta$ or need to replace a $1$ strategically.
Tool 3: Compound Angle Identities#
$$ \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta $$$$ \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta $$When to use it: When you see $\sin(\alpha + \beta)$ or $\cos(A - B)$ in the expression.
Tool 4: Double Angle Identities#
$$ \sin 2\theta = 2\sin\theta\cos\theta $$$$ \cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta $$When to use it: When you see $\sin 2\theta$, $\cos 2\theta$, or expressions like $2\sin\theta\cos\theta$.
Tool 5: Factoring#
- Difference of squares: $\cos^2\theta - \sin^2\theta = (\cos\theta - \sin\theta)(\cos\theta + \sin\theta)$
- Common factors: $\sin\theta\cos\theta + \sin\theta = \sin\theta(\cos\theta + 1)$
When to use it: When you need to simplify or cancel terms.
3. The Strategy (Step by Step)#
- Choose the complicated side (usually the LHS).
- Convert everything to $\sin$ and $\cos$ (remove all $\tan$).
- Look for compound/double angles and expand them.
- Find common denominators if there are fractions.
- Factor where possible.
- Use Pythagorean identity to simplify.
- Keep simplifying until you reach the other side.
4. Worked Examples#
Example 1: Basic Identity#
Prove that $\frac{\sin\theta}{\tan\theta} = \cos\theta$
LHS:
$$ = \frac{\sin\theta}{\frac{\sin\theta}{\cos\theta}} $$$$ = \sin\theta \times \frac{\cos\theta}{\sin\theta} $$$$ = \cos\theta $$$$ = \text{RHS} \checkmark $$Example 2: Using Pythagorean Identity#
Prove that $\frac{\sin^2\theta}{1 + \cos\theta} = 1 - \cos\theta$
LHS:
$$ = \frac{\sin^2\theta}{1 + \cos\theta} $$Replace $\sin^2\theta$ with $1 - \cos^2\theta$:
$$ = \frac{1 - \cos^2\theta}{1 + \cos\theta} $$Factor the numerator (difference of squares):
$$ = \frac{(1 - \cos\theta)(1 + \cos\theta)}{1 + \cos\theta} $$Cancel $(1 + \cos\theta)$:
$$ = 1 - \cos\theta $$$$ = \text{RHS} \checkmark $$Example 3: Using Double Angle#
Prove that $\frac{\sin 2A}{1 + \cos 2A} = \tan A$
LHS:
Expand the double angles:
- $\sin 2A = 2\sin A\cos A$
- $\cos 2A = 2\cos^2 A - 1$ (choose this form because $1 + (2\cos^2 A - 1) = 2\cos^2 A$ — the $1$s cancel!)
Cancel $2\cos A$:
$$ = \frac{\sin A}{\cos A} $$$$ = \tan A $$$$ = \text{RHS} \checkmark $$Example 4: Using Compound Angles#
Prove that $\cos(90° - x)\cos(90° - y) - \cos x\cos y = -\cos(x + y)$
LHS:
Apply reduction formulae: $\cos(90° - x) = \sin x$ and $\cos(90° - y) = \sin y$:
$$ = \sin x\sin y - \cos x\cos y $$Recognize the compound angle pattern (with a negative sign):
$$ = -(\cos x\cos y - \sin x\sin y) $$$$ = -\cos(x + y) $$$$ = \text{RHS} \checkmark $$Example 5: Fractions with Common Denominators#
Prove that $\frac{\cos\theta}{1 - \sin\theta} + \frac{\cos\theta}{1 + \sin\theta} = \frac{2}{\cos\theta}$
LHS:
Find common denominator $(1 - \sin\theta)(1 + \sin\theta)$:
$$ = \frac{\cos\theta(1 + \sin\theta) + \cos\theta(1 - \sin\theta)}{(1 - \sin\theta)(1 + \sin\theta)} $$Expand numerator:
$$ = \frac{\cos\theta + \cos\theta\sin\theta + \cos\theta - \cos\theta\sin\theta}{1 - \sin^2\theta} $$Simplify numerator ($\sin\theta$ terms cancel):
$$ = \frac{2\cos\theta}{1 - \sin^2\theta} $$Use Pythagorean identity: $1 - \sin^2\theta = \cos^2\theta$:
$$ = \frac{2\cos\theta}{\cos^2\theta} $$Cancel one $\cos\theta$:
$$ = \frac{2}{\cos\theta} $$$$ = \text{RHS} \checkmark $$5. Choosing the Right $\cos 2\theta$ Form#
This decision comes up in almost every double-angle proof. Here’s the cheat sheet:
| If you see… | Choose this form of $\cos 2\theta$ | Why |
|---|---|---|
| $1 + \cos 2\theta$ | $\cos 2\theta = 2\cos^2\theta - 1$ | Because $1 + (2\cos^2\theta - 1) = 2\cos^2\theta$ |
| $1 - \cos 2\theta$ | $\cos 2\theta = 1 - 2\sin^2\theta$ | Because $1 - (1 - 2\sin^2\theta) = 2\sin^2\theta$ |
| Only $\sin$ in the expression | $\cos 2\theta = 1 - 2\sin^2\theta$ | Keeps everything in terms of $\sin$ |
| Only $\cos$ in the expression | $\cos 2\theta = 2\cos^2\theta - 1$ | Keeps everything in terms of $\cos$ |
| Both $\sin$ and $\cos$ | $\cos 2\theta = \cos^2\theta - \sin^2\theta$ | The “original” form — factorises nicely |
🚨 Common Mistakes#
- Crossing the equals sign: You must ONLY work on one side. Writing “LHS = RHS, therefore $\sin\theta = ...$” is wrong — you’re proving, not solving.
- Forgetting to state the identity used: In your working, write the reason for each step (e.g., “double angle” or “Pythagorean identity”).
- Wrong $\cos 2\theta$ choice: If your proof gets messier instead of simpler after expanding, you probably chose the wrong form. Go back and try another.
- Not converting $\tan$: Leaving $\tan$ in the expression almost always makes things harder. Convert to $\frac{\sin}{\cos}$ first.
- Cancelling incorrectly: You can only cancel a factor if it appears in EVERY term of the numerator and denominator. $\frac{\sin\theta + 1}{\sin\theta}$ does NOT simplify to $1 + 1 = 2$.
💡 Pro Tip: Work Backwards (Secretly)#
If you’re stuck, look at the RHS and think about what operations would produce it. Then go back to the LHS and aim for those intermediate steps. Your final written answer must go LHS → RHS, but your rough work can explore from both directions.
⏮️ Double Angles | 🏠 Back to Trigonometry | ⏭️ General Solutions