Trigonometric Proof (Cambridge (CIE) IGCSE Additional Maths): Revision Note

Trigonometric proof

Proving trigonometric identities

• You can use trigonometric identities you already know to prove new identities

• Make sure you know the simple trigonometric identities and further trigonometric identities

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

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

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

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

• To prove an identity start on one side and proceed step by step until you get to the other side

  • e.g. to prove that ${\left(\sin \theta +\cos \theta \right)}^2=1+2\sin \theta \cos \theta$

  • Start with the left hand side, and expand

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

  • Simplify using the identity ${\sin }^2\theta +{\cos }^2\theta =1$

    • $1+2\sin \theta \cos \theta$

  • This is now equal to the expression on the right, so the statement has been proven

• Make sure you are confident handling fractions and fractions-within-fractions

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