Trigonometric Proof

How do I prove new trigonometric identities?

You can use trigonometric identities you already know to prove new identities
Make sure you know how to find all of the trig identities in the formula booklet
- The identity for tan, simple Pythagorean identity and the double angle identities for in and cos are in the SL section
  - $\tan θ = \frac{\sin θ}{\cos θ}$
  - $\cos^{2} θ + \sin^{2} θ = 1$
  - $\sin 2 θ = 2 \sin θ \cos θ$
  - $\cos 2 θ = \cos^{2} θ - \sin^{2} θ = 2 \cos^{2} θ - 1 = 1 - 2 \sin^{2} θ$
- The reciprocal trigonometric identities for sec and cosec, further Pythagorean identities, compound angle identities and the double angle formula for tan
  - $\sec θ = \frac{1}{\cos θ}$
  - $cosec θ = \frac{1}{\sin θ}$
  - $1 + \tan^{2} θ = \sec^{2} θ$
  - $1 + \cot^{2} θ = {cosec}^{2} θ$
  - $\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$
  - $\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$
  - $\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
  - $\tan 2 θ = \frac{2 \tan θ}{1 - \tan^{2} θ}$
- The identity for cot is not in the formula booklet, you will need to remember it
  - $\cot θ = \frac{1}{\tan θ} = \frac{\cos θ}{\sin θ}$
To prove an identity start on one side and proceed step by step until you get to the other side
- It is more common to start on the left hand side but you can start a proof from either end
- Occasionally it is easier to show that one side subtracted from the other is zero
- You should not work on both sides simultaneously

What should I look out for when proving new trigonometric identities?

Look for anything that could be a part of one of the above identities on either side

For example if you see $\sin 2 θ$ you can replace it with $2 \sin θ \cos θ$
If you see $2 \sin θ \cos θ$ you can replace it with $\sin 2 θ$

Look for ways of reducing the number of different trigonometric functions there are within the identity

For example if the identity contains tan θ, cot θ and cosec θ you could try

using the identities tan θ = 1/cot θ and 1 + cot² θ = cosec²θ to write it all in terms of cot θ
or rewriting it all in terms of sin θ and cos θ and simplifying

Often you may need to trial a few different methods before finding the correct one
Clever substitution into the compound angle formulae can be a useful tool for proving identities

For example rewriting $\cos \frac{θ}{2}$ as $\cos (θ - \frac{θ}{2})$ doesn’t change the ratio but could make an identity easier to prove

You will most likely need to be able to work with fractions and fractions-within-fractions
Always keep an eye on the 'target' expression – this can help suggest what identities to use

Examiner Tip

Don't forget that you can start a proof from either end – sometimes it might be easier to start from the left-hand side and sometimes it may be easier to start from the right-hand side
Make sure you use the formula booklet as all of the relevant trigonometric identities are given to you
Look out for special angles (0°, 90°, etc) as you may be able to quickly simplify or cancel parts of an expression (e.g. $\cos 90 ° = 0$ )

Worked example

Prove that $8 \cos^{4} θ - 8 \cos^{2} θ + 1 = \cos 4 θ$ .

3-8-1-ib-aa-hl-trig-proof-we-solution-

Trigonometric Proof (DP IB Maths: AA HL)

Revision Note

How do I prove new trigonometric identities?

What should I look out for when proving new trigonometric identities?

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1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.3.5 Compound Interest & Depreciation

1.4 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions

2.7.4 Roots of Polynomials

2.8 Inequalities

2.8.1 Solving Inequalities Graphically

2.8.2 Polynomial Inequalities

2.9 Further Functions & Graphs