Trig Identities and Proving Identities

CIE IGCSE Additional Mathematics· 10.4, 10.6 (2025-2027)· 25 min read

1. The 3 Pythagorean Trig Identities★★☆☆☆⏱ 5 min

📘 Definition

Pythagorean Trigonometric Identities (0606 approved)

The only three trig identities permitted for identity proofs in 0606:
1. $sin^{2} θ + cos^{2} θ = 1$
2. $sec^{2} θ = 1 + tan^{2} θ$
3. $cosec^{2} θ = 1 + cot^{2} θ$

Example:

For θ = 60°, $sin^{2} 60 + cos^{2} 60 = (\frac{3}{2})^{2} + (\frac{1}{2})^{2} = 1$ , satisfying the first identity.

📐 Worked Example

Derive $sec^{2} θ = 1 + tan^{2} θ$ from the identity $sin^{2} θ + cos^{2} θ = 1$ .

1
Divide all terms of $sin^{2} θ + cos^{2} θ = 1$ by $cos^{2} θ$ :
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: $tan θ = \frac{s i n θ}{c o s θ}$ and $sec θ = \frac{1}{c o s θ}$ :
4
$\tan^2 \theta + 1 = \sec^2 \theta$
5
Rearrange to match the required identity.

2. Core Strategies for Proving Trig Identities★★★☆☆⏱ 8 min

🚫 No Calculator

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 $sin θ$ and $cos θ$ first to eliminate $sec, cosec, tan, cot$ 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 $tan θ + cot θ = sec θ cosec θ$ .

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 $sin θ cos θ$ :
4
$LHS = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}$
5
Substitute $sin^{2} θ + cos^{2} θ = 1$ :
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.

3. Proofs with sec² and cosec² Identities★★★★☆⏱ 7 min

🚫 No Calculator

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 $sec^{4} θ - sec^{2} θ = tan^{4} θ + tan^{2} θ$ .

1
Start with the LHS, factor out the common term $sec^{2} θ$ :
2
$LHS = \sec^2 \theta (\sec^2 \theta - 1)$
3
Use the rearranged identity $sec^{2} θ - 1 = tan^{2} θ$ to substitute:
4
$LHS = \sec^2 \theta \times \tan^2 \theta$
5
Rewrite $sec^{2} θ$ using the identity $sec^{2} θ = 1 + tan^{2} θ$ :
6
$LHS = (1 + \tan^2 \theta) \tan^2 \theta = \tan^2 \theta + \tan^4 \theta$
7
Rearrange to match the RHS: $tan^{4} θ + tan^{2} θ = R H S$ . The identity is proven.

✓ Quick check

Which substitution correctly simplifies $1 - cosec^{2} θ$ ?
- $- cot^{2} θ$
- $cot^{2} θ$
- $- tan^{2} θ$
- $sec^{2} θ$

4. Structuring Proofs for Full Exam Marks★★★☆☆⏱ 5 min

📐 Worked Example

A 3-mark exam question asks: Prove $cosec θ - sin θ = cot θ cos θ$ . 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 $sin θ$ :
4
$LHS = \frac{1 - \sin^2 \theta}{\sin \theta}$
5
Substitute $1 - sin^{2} θ = cos^{2} θ$ :
6
$LHS = \frac{\cos^2 \theta}{\sin \theta} = \frac{\cos \theta}{\sin \theta} \times \cos \theta$
7
Substitute $\frac{c o s θ}{s i n θ} = cot θ$ to get $cot θ cos θ = R H S$ . The identity is proven.

5. Common Pitfalls

Wrong move:

Using compound-angle, double-angle or other out-of-scope identities in proofs

Why:

These are not permitted per the 0606 syllabus, markers will deduct marks even if your proof works

Correct move:

Only use the three Pythagorean identities and basic trig ratio definitions

Wrong move:

Moving terms across the equals sign (e.g., subtracting RHS from LHS) in the final proof

Why:

This assumes the identity is true before you prove it, which is circular logic

Correct move:

Manipulate only one side, or both sides separately, to reach the same expression

Wrong move:

Interpreting $sin^{2} θ$ as $sin (θ^{2})$ instead of $(sin θ)^{2}$

Why:

Misinterpreting notation leads to incorrect algebraic manipulation

Correct move:

Treat squared trig terms as standard algebraic squared terms when factoring or simplifying

Wrong move:

Cancelling $sin θ$ or other trig terms from both sides of an expression without checking if it is zero

Why:

Division by zero is undefined, leading to invalid steps

Correct move:

Factor terms out instead of cancelling, or note the identity holds for all values where expressions are defined

Wrong move:

Misremembering the cosec identity as $cosec^{2} θ + 1 = cot^{2} θ$

Why:

Swapping terms leads to incorrect substitutions and wrong results

Correct move:

Recall that 'co' functions pair together: sec² pairs with tan², cosec² pairs with cot²

6. Quick Reference Cheatsheet

Identity	Rearranged Forms	Use Case
$sin^{2} θ + cos^{2} θ = 1$	$1 - sin^{2} θ = cos^{2} θ$ , $1 - cos^{2} θ = sin^{2} θ$	Simplify squared sin/cos terms, combine fractions
$sec^{2} θ = 1 + tan^{2} θ$	$sec^{2} θ - 1 = tan^{2} θ$ , $sec^{2} θ - tan^{2} θ = 1$	Substitute for squared sec/tan terms, factor expressions
$cosec^{2} θ = 1 + cot^{2} θ$	$cosec^{2} θ - 1 = cot^{2} θ$ , $cosec^{2} θ - cot^{2} θ = 1$	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

7. Frequently Asked

Do I need to memorize the three Pythagorean identities for the exam?

Yes, while select identities may be listed in the formula booklet, memorizing them will drastically speed up your work on both non-calculator Paper 1 and calculator Paper 2, and reduce errors from looking up references.

Can I use compound angle or double angle formulas in my identity proofs?

No, per the 0606 syllabus, only the three Pythagorean identities and basic trig ratio definitions are permitted for identity proofs. Using out-of-scope formulas will result in lost marks even if your final result is correct.

Going deeper

syllabus_documentCIE 0606 2025-2027 SyllabusSee section 10 for trigonometry requirements

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.