# Trigonometric Identities

Trigonometric identities are identities in mathematics that involve trigonometric functions such as sin(x), cos(x) and tan(x). Identities, as opposed to equations, are statements where the left hand side is equivalent to the right hand side. A $\equiv$ symbol, which means ‘equivalent’, is used instead of the $=$ which means ‘equals’. Equations can be solved to find values of x for instance. An identity, however, cannot be solved, no additional information has been given but they can be used to solve equations. See Trigonometric Equations.

## AS-Level Trigonometric Identities

These are the formulae that you should memorise for the first year of your A-Level in Mathematics:

### Fundamental Formulae

$\frac{\sin(\theta)}{\cos(\theta)}\equiv\tan(\theta)$
$\cos^2(\theta)+\sin^2(\theta)\equiv 1$

$\sin(-\theta)=-\sin(\theta)$
$\cos(-\theta)=\cos(\theta)$
$\sin(90^\circ-\theta)=\cos(\theta)$
$\cos(90^\circ-\theta)=\sin(\theta)$
$\sin(180^\circ-\theta)=\sin(\theta)$
$\cos(180^\circ-\theta)=-\cos(\theta)$
These identities can be seen by performing trigonometric graph transformations.

## A-Level Trigonometric Identities

These are the formulae that you should memorise for the second year of your A-Level in Mathematics:

### Double Angle Formulae

$\sin(2\theta)\equiv2\sin(\theta)\cos(\theta)$
$\cos(2\theta)\equiv\cos(\theta)^2-\sin^2(\theta)$
$\cos(2\theta)\equiv2\cos^2(\theta)-1$
$\cos(2\theta)\equiv1-2\sin^2(\theta)$

### Compound Angle Formulae

$\sin(\alpha\pm\beta)\equiv\sin(\alpha)\cos(\beta)\pm\cos(\alpha)\sin(\beta)$
$\cos(\alpha\pm\beta)\equiv\sin(\alpha)\sin(\beta)\mp\cos(\alpha)\cos(\beta)$
$\tan(\alpha\pm\beta)\equiv\frac{\tan(\alpha)\pm\tan(\beta)}{1\mp\tan(\alpha)\tan(\beta)}$

For more trigonometric identities visit Wikipedia Trig. Identities.

See other Things to Remember.

## Example 1

Simplify

$\frac{\sqrt{1-\cos^2(x)}}{\cos(x)}$

## Example 2

Show that $\frac{\sin^4(\theta)-\cos^4(\theta)}{\cos^2(\theta)}=\tan^2(\theta)-1$.

## Example 3

Given that $\theta$ is obtuse and $\cos(\theta)=-\frac{2}{5}$, find the value of $\tan(\theta)$.

## Example 4

Given that $x=2\cos(\alpha)$ and $y=3\sin(\alpha)$, find the value of $9x^2+4y^2$.

### Past Trigonometric Exam Questions

We have collated past exam questions on trigonometry so that you may focus your concentration on this particular subject (answers on the back pages). Alternatively, visit our Questions by Topic page to see which topics you can focus on.

### Pure Maths Practice Papers

Are you ready to test your knowledge on all of the Pure Maths topics so far? If so…