# Trig Identities and Proving Identities
> CIE IGCSE Additional Mathematics · 0606
> Source: https://www.owlsprep.com/study/cie-0606-u10-trig-identities-and-proving-identities/
This guide covers the three Pythagorean trig identities for CIE IGCSE Additional Mathematics 0606, plus step-by-step strategies to prove trigonometric relationships using only these identities and basic ratio definitions.
**Prerequisites:** Reciprocal trigonometric ratios (sec, cosec, cot); Basic sine, cosine, tangent ratio definitions
## Learning objectives
- Recall the three Pythagorean trigonometric identities for 0606
- Apply ratio definitions and Pythagorean identities to prove trigonometric relationships
- Avoid common logical and algebraic errors in identity proofs
- Structure exam answers for full marks on identity proof questions
## The 3 Pythagorean Trig Identities
**Pythagorean Trigonometric Identities (0606 approved)** — The only three trig identities permitted for identity proofs in 0606:
1. $\sin^2 \theta + \cos^2 \theta = 1$
2. $\sec^2 \theta = 1 + \tan^2 \theta$
3. $\text{cosec}^2 \theta = 1 + \cot^2 \theta$
*Example:* For θ = 60°, $\sin^2 60 + \cos^2 60 = (\frac{\sqrt{3}}{2})^2 + (\frac{1}{2})^2 = 1$, satisfying the first identity.
**Worked example:** Derive $\sec^2 \theta = 1 + \tan^2 \theta$ from the identity $\sin^2 \theta + \cos^2 \theta = 1$.
1. Divide all terms of $\sin^2 \theta + \cos^2 \theta = 1$ by $\cos^2 \theta$:
2. $$\frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta}$$
3. Substitute using ratio definitions: $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\sec \theta = \frac{1}{\cos \theta}$:
4. $$\tan^2 \theta + 1 = \sec^2 \theta$$
5. Rearrange to match the required identity.
**Summary**
- All three identities are derived from the Pythagorean theorem for right triangles
- You can rearrange each identity to substitute for squared terms in proofs
- No other trig identities are allowed for 0606 identity proofs
## Core Strategies for Proving Trig Identities
When proving trig identities, always start from the more complex side of the equality and simplify it to match the simpler side. You may manipulate both sides separately to reach a common intermediate expression, but never move terms across the equals sign in your final proof.
- Express all terms in terms of $\sin \theta$ and $\cos \theta$ first to eliminate $\sec, \text{cosec}, \tan, \cot$ if possible
- Look for opportunities to substitute one of the three Pythagorean identities to simplify squared terms
- Factor algebraic expressions (e.g., difference of squares) to cancel terms
- Combine fractions using a common denominator to simplify rational trig expressions
**Worked example:** Prove that $\tan \theta + \cot \theta = \sec \theta \text{cosec} \theta$.
1. Start with the left-hand side (LHS), rewrite tan and cot as ratios of sin and cos:
2. $$LHS = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}$$
3. Combine the fractions using common denominator $\sin \theta \cos \theta$:
4. $$LHS = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}$$
5. Substitute $\sin^2 \theta + \cos^2 \theta = 1$:
6. $$LHS = \frac{1}{\sin \theta \cos \theta}$$
7. Split into product of reciprocals to match the right-hand side (RHS):
8. $$LHS = \frac{1}{\cos \theta} \times \frac{1}{\sin \theta} = \sec \theta \text{cosec} \theta = RHS$$
9. The identity is proven.
> **tip**
>
> Always label which side you are working on (LHS/RHS) in your exam answer to make your logic clear to the marker.
*Calculator:* forbidden
## Proofs with sec² and cosec² Identities
Most higher-mark exam proof questions test use of the sec² and cosec² identities. These often require rearranging the identities to substitute for squared terms, rather than using the sin²/cos² identity directly.
**Worked example:** Prove that $\sec^4 \theta - \sec^2 \theta = \tan^4 \theta + \tan^2 \theta$.
1. Start with the LHS, factor out the common term $\sec^2 \theta$:
2. $$LHS = \sec^2 \theta (\sec^2 \theta - 1)$$
3. Use the rearranged identity $\sec^2 \theta - 1 = \tan^2 \theta$ to substitute:
4. $$LHS = \sec^2 \theta \times \tan^2 \theta$$
5. Rewrite $\sec^2 \theta$ using the identity $\sec^2 \theta = 1 + \tan^2 \theta$:
6. $$LHS = (1 + \tan^2 \theta) \tan^2 \theta = \tan^2 \theta + \tan^4 \theta$$
7. Rearrange to match the RHS: $\tan^4 \theta + \tan^2 \theta = RHS$. The identity is proven.
**Check your understanding**
1. Which substitution correctly simplifies $1 - \text{cosec}^2 \theta$?
- $-\cot^2 \theta$
- $\cot^2 \theta$
- $-\tan^2 \theta$
- $\sec^2 \theta$
*Answer:* $-\cot^2 \theta$
*Why:* Rearranging the identity $\text{cosec}^2 \theta = 1 + \cot^2 \theta$ gives $1 - \text{cosec}^2 \theta = -\cot^2 \theta$.
*Calculator:* forbidden
## Structuring Proofs for Full Exam Marks
**Exam command terms**
CIE markers expect clear, logical steps for identity proof questions. Key command terms you will encounter include:
- **Prove that** — Show full logical steps from one side to the other, no gaps, clearly label LHS/RHS *(Prove that $\sin^2 \theta (1 + \cot^2 \theta) = 1$)*
- **Show that** — Simplify an expression to reach a given result, show all intermediate steps *(Show that $(\sec \theta - 1)(\sec \theta + 1) = \tan^2 \theta$)*
**Worked example:** A 3-mark exam question asks: Prove $\text{cosec} \theta - \sin \theta = \cot \theta \cos \theta$. Show the full worked solution for full marks.
1. Start with LHS, rewrite cosec as reciprocal of sin:
2. $$LHS = \frac{1}{\sin \theta} - \sin \theta$$
3. Combine terms over common denominator $\sin \theta$:
4. $$LHS = \frac{1 - \sin^2 \theta}{\sin \theta}$$
5. Substitute $1 - \sin^2 \theta = \cos^2 \theta$:
6. $$LHS = \frac{\cos^2 \theta}{\sin \theta} = \frac{\cos \theta}{\sin \theta} \times \cos \theta$$
7. Substitute $\frac{\cos \theta}{\sin \theta} = \cot \theta$ to get $\cot \theta \cos \theta = RHS$. The identity is proven.
> **warning**
>
> You will lose marks if you skip substitution steps or use out-of-scope identities, even if your final result is correct.
## Common pitfalls
- **Wrong:** Using compound-angle, double-angle or other out-of-scope identities in proofs
- Why it fails: These are not permitted per the 0606 syllabus, markers will deduct marks even if your proof works
- Correct: Only use the three Pythagorean identities and basic trig ratio definitions
- **Wrong:** Moving terms across the equals sign (e.g., subtracting RHS from LHS) in the final proof
- Why it fails: This assumes the identity is true before you prove it, which is circular logic
- Correct: Manipulate only one side, or both sides separately, to reach the same expression
- **Wrong:** Interpreting $\sin^2 \theta$ as $\sin(\theta^2)$ instead of $(\sin \theta)^2$
- Why it fails: Misinterpreting notation leads to incorrect algebraic manipulation
- Correct: Treat squared trig terms as standard algebraic squared terms when factoring or simplifying
- **Wrong:** Cancelling $\sin \theta$ or other trig terms from both sides of an expression without checking if it is zero
- Why it fails: Division by zero is undefined, leading to invalid steps
- Correct: Factor terms out instead of cancelling, or note the identity holds for all values where expressions are defined
- **Wrong:** Misremembering the cosec identity as $\text{cosec}^2 \theta + 1 = \cot^2 \theta$
- Why it fails: Swapping terms leads to incorrect substitutions and wrong results
- Correct: Recall that 'co' functions pair together: sec² pairs with tan², cosec² pairs with cot²
## Cheatsheet
| Identity | Rearranged Forms | Use Case |
| --- | --- | --- |
| $\sin^2 \theta + \cos^2 \theta = 1$ | $1 - \sin^2 \theta = \cos^2 \theta$, $1 - \cos^2 \theta = \sin^2 \theta$ | Simplify squared sin/cos terms, combine fractions |
| $\sec^2 \theta = 1 + \tan^2 \theta$ | $\sec^2 \theta - 1 = \tan^2 \theta$, $\sec^2 \theta - \tan^2 \theta = 1$ | Substitute for squared sec/tan terms, factor expressions |
| $\text{cosec}^2 \theta = 1 + \cot^2 \theta$ | $\text{cosec}^2 \theta - 1 = \cot^2 \theta$, $\text{cosec}^2 \theta - \cot^2 \theta = 1$ | Substitute for squared cosec/cot terms, simplify rational expressions |
| Proof Strategy | | Start from complex side, express all terms in sin/cos first, look for Pythagorean substitutions |
## What's next
Now that you have mastered proving trig identities using the three Pythagorean identities, you are ready to apply these skills to solve trigonometric equations, the next key topic in the CIE IGCSE Additional Mathematics 0606 trigonometry unit. You will also use these identities when differentiating and integrating trigonometric functions later in the course, so it is critical to memorize the three identities and practice proof questions regularly to build speed and accuracy for both Paper 1 (non-calculator) and Paper 2. Make sure you work through past paper identity proof questions to familiarize yourself with common exam formats and marker expectations.
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