Trigonometric Proof (DP IB Analysis & Approaches (AA)): Revision Note

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

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  • The reciprocal trigonometric identities for sec and cosec, further Pythagorean identities, compound angle identities and the double angle formula for tan

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  • The identity for cot is not in the formula booklet, you will need to remember it

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• 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  you can replace it with 

  • If you see you can replace it with 

• 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 = cosec² θ 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 as 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