Trigonometric Proof

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

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

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

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

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. $\cos 90 ° = 0$ )

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

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Trigonometric Proof (DP IB Maths: AA HL)