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

Trigonometric proof

What identities might I have to use during a proof?

• You are given the following identities in the SL section of the formula booklet

  • Identity for $\tan \theta$

    • $\tan \theta =\frac{\sin \theta }{\cos \theta }$

  • Pythagorean identity

    • ${\cos }^2\theta +{\sin }^2\theta =1$

  • Double angle identities

    • $\sin 2\theta =2\sin \theta \cos \theta$

    • $\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta =2{\cos }^2\theta -1=1-2{\sin }^2\theta$

• You are given the following identities in the HL section of the formula booklet

  • Reciprocal trigonometric identities

    • $\sec \theta =\frac{1}{\cos \theta }$

    • $\mathrm{cosec} \theta =\frac{1}{\sin \theta }$

  • Pythagorean identities

    • $1+{\tan }^2\theta ={\sec }^2\theta$

    • $1+{\cot }^2\theta ={\mathrm{cosec}}^2\theta$

  • Compound angle identities

    • $\sin \left(A\pm B\right)=\sin A\cos B \pm \cos A\sin B$

    • $\cos \left(A\pm B\right)=\cos A\cos B \mp \sin A\sin B$

    • $\tan \left(A\pm B\right)=\frac{\tan A \pm \tan B}{1\mp \tan A\tan B}$

  • Double angle identity for tan

    • $\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }$

• You are not given the following identities and need to remember them

  • Reciprocal trigonometric identity for cot

    • $\cot \theta =\frac{1}{\tan \theta }$

  • Identity for $\cot \theta$

    • $\cot \theta =\frac{\cos \theta }{\sin \theta }$

How do I prove an identity?

• To prove an identity:

  • Select one side to start on

    • It is more common to start on the left-hand side

    • However, you can start on the right-hand side

  • Apply relevant identities to turn that expression into the one on the other side

What should I look out for when proving trigonometric identities?

• Check to see if any of the angles are double or half any of the others

  • You can use the double angle identities

    • e.g. you can replace $\sin 6\theta$ with $2\sin 3\theta \cos 3\theta$

    • e.g. you can replace $\cos 4\theta$with $1-2{\sin }^{2}2\theta$

    • e.g. you can replace $\sin \theta \cos \theta$ with $\frac{1}{2}\sin 2\theta$

• Check to see if any of the terms have an even power

  • You can use the Pythagorean identities

    • e.g. you can replace ${\sec }^4\theta$ with ${\left(1+{\tan }^2\theta \right)}^2$

• Check to see if any terms can cancel

  • e.g. you can replace $\cos 4\theta$ with $2{\cos }^{2}2\theta -1$ to cancel the 1 in the expression$1+\cos 4\theta$

• Combine any fractions

  • e.g. rewrite $\frac{1}{\cos \theta }+\frac{1}{\sin \theta }$ as $\frac{\sin \theta +\cos \theta }{\cos \theta \sin \theta }$

• Always keep an eye on the 'target' expression – this can help suggest what identities to use