3.8.1 Trigonometric Proof

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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