CAIE A-Level Mathematics 3.3.2 Trigonometric Identities

Practise proving and applying reciprocal, compound-angle, double-angle and R-form identities to simplify expressions and solve equations across restricted intervals.

Exam points

rewrite sec, cosec and cot or expand compound and double angles in sine and cosine
transform a sin x + b cos x into an R form using R² = a² + b² and the correct α
use the derived identity to solve for every angle in range and reject extraneous branches

[Maximum number: 6]

(a)

Show that the equation $\tan^3x+2\tan2x-\tan x=0$ may be expressed as
$\tan^4x-2\tan^2x-3=0$
for $\tan x\ne0$ .

[ 3 ]

Use the double-angle formula:
$\tan2x=\frac{2\tan x}{1-\tan^2x}.$
M1

For example,
$\tan^3x+\frac{4\tan x}{1-\tan^2x}-\tan x=0.$
Multiplying by $1-\tan^2x$ gives a correct equation in $\tan x$ , such as
$\tan^3x-\tan^5x+4\tan x-\tan x+\tan^3x=0.$
A1

Since $\tan x\ne0$ , reduce correctly to
$\tan^4x-2\tan^2x-3=0.$
A1

(b)

Hence solve the equation $\tan^3 2\theta+2\tan4\theta-\tan2\theta=0$ for $0<\theta<\pi$ . Give your answers in exact form.

[ 3 ]

Using part (a) with $x=2\theta$ :
$\tan^4 2\theta-2\tan^2 2\theta-3=0.$
Thus
$\tan^2 2\theta=3,\qquad \tan2\theta=\pm\sqrt3.$
M1

For $0<\theta<\pi$ ,
$\theta=\frac{\pi}{6},\quad \frac{\pi}{3},\quad \frac{2\pi}{3},\quad \frac{5\pi}{6}.$
A1, A1

Exact answers only; no others in the interval.

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3.3.2—Trig identities question 1

Question (a)

Question (b)