A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 7: Trigonometry and modelling · Lesson 5

Proving trigonometric identities

115 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-proving-trig-identities/

Date

Do Now

Quick questions to start.

A1
Last lesson
Write $cos 2 x$ in terms of $sin x$ only.
[2 marks]
Answer
A2
From chapter 6
Show that $\frac{sin ^{2} x}{1 - cos x} \equiv 1 + cos x$ .
[2 marks]
A3
Needed today
Show that $cos (x - 4 5^{\circ}) \equiv \frac{2}{2} (cos x + sin x)$ .
[2 marks]
A4
Today's idea
Show that $tan x + cot x \equiv \frac{1}{sin x cos x}$ .
[3 marks]

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Prove that $\frac{sin 2 x}{1 + cos 2 x} \equiv tan x$ .

Hint Choose the form of cos 2x that makes the 1 disappear: 1 + cos 2x = 2 cos² x.

B1
Prove that $\frac{1 - cos 2 x}{1 + cos 2 x} \equiv tan^{2} x$ .
[4 marks]

Example 2

Prove that $cos^{4} x - sin^{4} x \equiv cos 2 x$ .

Hint It is a difference of two squares.

B2
Prove that $sin^{3} x cos x - sin x cos^{3} x \equiv - \frac{1}{4} sin 4 x$ .
[4 marks]

Example 3

Prove that $\frac{sin ( A + B )}{cos A cos B} \equiv tan A + tan B$ .

Hint Expand the top, then split the fraction into two.

B3
Prove that $\frac{cos ( A - B )}{sin A cos B} \equiv cot A + tan B$ .
[4 marks]

Example 4

Prove that $sin 3 x \equiv 3 sin x - 4 sin^{3} x$ .

Hint Write 3x as 2x + x and use the addition formula. Then change every cos² x to 1 − sin² x.

B4
Prove that $tan 3 x \equiv \frac{3 tan x - tan ^{3} x}{1 - 3 tan ^{2} x}$ .
[5 marks]

Example 5

Prove that $cosec x - sin x \equiv cos x cot x$ .

Hint Write cosec x as 1/sin x and put everything over sin x.

B5
Prove that $\frac{1 + tan ^{2} x}{1 - tan ^{2} x} \equiv sec 2 x$ .
[4 marks]

Practice

A-Level

C1
Mo is asked to prove that $tan x + cot x \equiv 2 cosec 2 x$ . He writes $tan x + cot x = \frac{1}{sin x cos x} = \frac{1}{sin 2 x} = cosec 2 x$ , so he says the identity is wrong. Find his mistake and complete the proof.
[4 marks]

C2
Prove that $\frac{sin 2 x - sin x}{1 + cos 2 x - cos x} \equiv tan x$ .
[5 marks]
C3
Prove that $cos (x + y) cos (x - y) \equiv cos^{2} x - sin^{2} y$ .
[4 marks]
C4
(a)
Prove that $8 sin^{2} x cos^{2} x \equiv 1 - cos 4 x$ .
[3 marks]
(b)
Hence find the exact value of $sin^{2} \frac{π}{8} cos^{2} \frac{π}{8}$ .
[2 marks]
Answer
Total for C4: 5 marks
C5
(a)
Prove that $cot x - tan x \equiv 2 cot 2 x$ .
[3 marks]
(b)
Hence find the exact value of $cot \frac{π}{8} - tan \frac{π}{8}$ .
[1 mark]
Answer
Total for C5: 4 marks

Exam-style questions

Edexcel exam style

D1
(a)
Prove that $\frac{cos 2 x}{cos x + sin x} \equiv cos x - sin x$ , for $cos x + sin x \neq = 0$ .
[3 marks]
(b)
Express $cos x - sin x$ in the form $R cos (x + α)$ , where $R > 0$ and $0^{\circ} < α < 9 0^{\circ}$ .
[3 marks]
Answer
(c)
Hence solve $\frac{cos 2 x}{cos x + sin x} = 1$ for $0^{\circ} \leq x < 36 0^{\circ}$ .
[3 marks]
Answer
(d)
State the values of $x$ in $0^{\circ} \leq x < 36 0^{\circ}$ for which $\frac{cos 2 x}{cos x + sin x}$ is not defined.
[2 marks]
Answer
Total for D1: 11 marks

D2
Jo writes: " $cos 4 x = cos (2 x + 2 x) = 2 cos^{2} 2 x - 1$ , and $cos^{2} 2 x = 2 cos^{2} x - 1$ , so $cos 4 x \equiv 4 cos^{2} x - 3$ ."
Jo says she has proved an identity for $cos 4 x$ .
(a)
Show that Jo is wrong.
[2 marks]
(b)
Prove that $cos 4 x \equiv 8 cos^{4} x - 8 cos^{2} x + 1$ .
[3 marks]
(c)
Hence solve $8 cos^{4} x - 8 cos^{2} x + 1 = \frac{1}{2}$ for $0^{\circ} \leq x < 9 0^{\circ}$ .
[4 marks]
Answer
Total for D2: 9 marks
D3
Prove that $\frac{sin 3 x}{sin x} - \frac{cos 3 x}{cos x} \equiv 2$ , for $sin x cos x \neq = 0$ .
[6 marks]
D4
Let $t = tan \frac{x}{2}$ .
(a)
Show that $sin x = \frac{2 t}{1 + t ^{2}}$ .
[3 marks]
(b)
Show that $cos x = \frac{1 - t ^{2}}{1 + t ^{2}}$ .
[2 marks]
(c)
Use the substitution to solve $2 sin x - cos x = 1$ for $0^{\circ} \leq x < 36 0^{\circ}$ . Give your answers to 1 decimal place. Check for any solution the substitution cannot find.
[5 marks]
Answer
Total for D4: 10 marks
D5
(a)
Prove that $cos (x + 6 0^{\circ}) + cos (x - 6 0^{\circ}) \equiv cos x$ .
[3 marks]
(b)
Prove that $sin (x + 6 0^{\circ}) - sin (x - 6 0^{\circ}) \equiv 3 cos x$ .
[3 marks]
(c)
Hence solve $cos (x + 6 0^{\circ}) + cos (x - 6 0^{\circ}) = sin (x + 6 0^{\circ}) - sin (x - 6 0^{\circ})$ for $0^{\circ} \leq x < 36 0^{\circ}$ .
[3 marks]
Answer
Total for D5: 9 marks
D6
(a)
Prove that $sec^{2} x + cosec^{2} x \equiv sec^{2} x cosec^{2} x$ .
[3 marks]
(b)
Hence prove that $sec^{2} x + cosec^{2} x \equiv 4 cosec^{2} 2 x$ .
[3 marks]
(c)
Solve $sec^{2} x + cosec^{2} x = \frac{16}{3}$ for $0 < x < \frac{π}{2}$ , giving exact answers.
[4 marks]
Answer
Total for D6: 10 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Prove that $cos x cos 2 x cos 4 x \equiv \frac{sin 8 x}{8 sin x}$ , for $sin x \neq = 0$ . Hence find the exact value of $cos \frac{π}{9} cos \frac{2 π}{9} cos \frac{4 π}{9}$ .
[8 marks]
Hint 1 · What to try
Multiply the left-hand side by $sin x$ . What does $sin x cos x$ make?
Hint 2 · The first line
Each step doubles the angle: $sin x cos x = \frac{1}{2} sin 2 x$ , then $sin 2 x cos 2 x = \frac{1}{2} sin 4 x$ , and so on.
Hint 3 · The full method
At $x = \frac{π}{9}$ you need $sin \frac{8 π}{9}$ . The angles $\frac{8 π}{9}$ and $\frac{π}{9}$ add to $π$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can start from one side of an identity and work, one line at a time, to the other side.
I can choose between the three forms of cos 2x and use sin 2x = 2 sin x cos x in either direction.
I can use a proved identity to find an exact value or to solve an equation.
I can give a counterexample to show that a statement is not an identity.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers