Trigonometric Identities (Cambridge (CIE) O Level Additional Maths): Revision Note

Simple trig identities

What is a trigonometric identity?

• Trigonometric identities are statements that are true for all values of $x$ or $\theta$

• They are used to help simplify trigonometric equations before solving them

• Sometimes you may see identities written with the symbol ≡

  • This means 'identical to'

What trigonometric identities do I need to know?

• The two trigonometric identities you must know are

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

    • This is the identity for tan θ

    • This formula does not appear in the list of formulae

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

    • This is the Pythagorean identity

    • Note that the notation $\sin {}^2\theta$ is the same as ${\left(\sin \theta \right)}^2$

    • This formula appears in the list of formulae

• Rearranging the second identity often makes it easier to work with

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

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

Where do the trigonometric identities come from?

• You do not need to know the proof for these identities but it is a good idea to know where they come from

• From SOHCAHTOA we know that

  • $\sin \theta =\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}=\frac{O}{H}$

  • $\cos \theta =\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}=\frac{A}{H}$

  • $\tan \theta =\frac{\mathrm{opposite}}{\mathrm{adjacent}}=\frac{O}{A}$

• The identity for $\tan \theta$ can be seen by diving $\sin \theta$ by $\cos \theta$?

  • $\frac{\sin \theta }{\cos \theta }=\frac{\frac{O}{H}}{\frac{A}{H}}=\frac{O}{A}=\tan \theta$

• This can also be seen from the unit circle by considering a right-triangle with a hypotenuse of 1

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

• The Pythagorean identity can be seen by considering a right-triangle with a hypotenuse of 1

  • Then (opposite)² + (adjacent)² = 1

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

• Considering the equation of the unit circle also shows the Pythagorean identity

  • The equation of the unit circle is $x^2+y^2=1$

  • The coordinates on the unit circle are $(\cos \theta , \sin \theta )$

  • Therefore the equation of the unit circle could be written ${\cos }^2\theta +{\sin }^2\theta =1$

How are the trigonometric identities used?

• Most commonly trigonometric identities are used to change an equation into a form that allows it to be solved

• They can also be used to prove further identities

Further trig identities

What are the identities linking tan, sec, cot, and cosec?

• Aside from the Pythagorean identity sin²x + cos²x = 1 there are two further Pythagorean identities you will need to learn

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

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

  • Both can be found in the list of formulae

• Both of these identities can be derived from sin²x + cos²x = 1 

  • To derive the identity for sec²x divide sin²x + cos²x = 1 by cos²x

  • To derive the identity for cosec²x divide sin²x + cos²x = 1 by sin²x

How do I prove new trigonometric identities?

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

• 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

  • For example if you see ${\sin }^2\theta$ you can replace it with $1-{\cos }^2\theta$

• 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

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