Further trigonometric identitiesEdexcel International A Level Maths: Revision notes

Section 1

Where the identities come from

Start from $\sin^2\theta+\cos^2\theta\equiv1$ . Dividing every term by $\cos^2\theta$ gives $\tan^2\theta+1\equiv\sec^2\theta$ : $\sec^2\theta\equiv1+\tan^2\theta.$ Dividing every term by $\sin^2\theta$ gives $1+\cot^2\theta\equiv\operatorname{cosec}^2\theta$ : $\operatorname{cosec}^2\theta\equiv1+\cot^2\theta.$ An identity is true for every value of $\theta$ where both sides are defined, unlike an equation, which is true only for certain values. The first identity holds when $\cos\theta\ne0$ , the second when $\sin\theta\ne0$ .

Key termsidentity

Exam tip

Rearranged forms are just as useful: $\tan^2\theta\equiv\sec^2\theta-1$ and $\cot^2\theta\equiv\operatorname{cosec}^2\theta-1$ .

Section 2

Finding exact values

If one trig ratio is known, the identities give the reciprocal functions without finding the angle. For acute $\theta$ with $\tan\theta=\frac{5}{12}$ : $\sec^2\theta=1+\frac{25}{144}=\frac{169}{144}$ , so $\sec\theta=\frac{13}{12}$ . Also $\cot\theta=\frac{12}{5}$ , so $\operatorname{cosec}^2\theta=1+\frac{144}{25}=\frac{169}{25}$ and $\operatorname{cosec}\,\theta=\frac{13}{5}$ . Taking a square root gives $\pm$ , so use the quadrant of $\theta$ to choose the sign.

Common mistake

Forgetting the sign after a square root. For an obtuse or reflex angle, $\sec\theta$ or $\operatorname{cosec}\,\theta$ may be negative.

Section 3

Proving identities

To prove an identity, start with one side (usually the more complicated) and manipulate it until it equals the other side. Write everything in terms of sine and cosine or use the identities to swap squares. Example. Prove $\dfrac{\tan^2\theta}{\sec\theta+1}\equiv\sec\theta-1$ . $\dfrac{\tan^2\theta}{\sec\theta+1}=\dfrac{\sec^2\theta-1}{\sec\theta+1}=\dfrac{(\sec\theta-1)(\sec\theta+1)}{\sec\theta+1}=\sec\theta-1$ . Here the difference of two squares does the work.

Key termsdifference of two squares

Common mistake

Treating the identity as an equation and doing something to both sides. Work on one side only until it matches the other.

Section 4

Solving equations

When an equation mixes a squared reciprocal function with a first power (or with another function), use an identity to get a quadratic in a single function. Example. Solve $\sec^2\theta=4\tan\theta-2$ for $0\le\theta<360^{\circ}$ . $1+\tan^2\theta=4\tan\theta-2\Rightarrow\tan^2\theta-4\tan\theta+3=0\Rightarrow(\tan\theta-1)(\tan\theta-3)=0$ . $\tan\theta=1$ : $\theta=45^{\circ},225^{\circ}$ . $\tan\theta=3$ : $\theta=71.6^{\circ},251.6^{\circ}$ . For an equation in $\operatorname{cosec}^2\theta$ and $\cot\theta$ , replace $\operatorname{cosec}^2\theta$ with $1+\cot^2\theta$ , solve for $\cot\theta$ , then use $\tan\theta=\frac{1}{\cot\theta}$ to find angles.

Exam tip

Make sure the quadratic is in ONE function before factorising: replace the squared term, not the first-power term.

Common mistake

Dividing both sides of an equation by a trig function, which loses solutions. Factorise instead.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Further trigonometric identities

The angle $\theta$ is acute and $\tan\theta=\frac{5}{12}$ .
Find the exact value of $\sec\theta-\tan\theta$ .2 marks
Throughout, $x$ is any angle for which the expressions are defined.
Show that $\dfrac{\sec^2x-1}{\sec^2x}\equiv\sin^2x$ .2 marks
Angles are measured in degrees and $0\le\theta<360$ .
Given that $\sec^2\theta=4\tan\theta-2$ , find the two possible values of $\tan\theta$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).