A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 6: Trigonometric functions · Lesson 2

Identities with sec, cosec and cot

117 marks

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Quick questions to start.

A1
Needed today
Show that $\frac{1 - cos ^{2} x}{sin x cos x} \equiv tan x$ .
[3 marks]
A2
Needed today
Solve $2 sin^{2} x - sin x - 1 = 0$ for $0^{\circ} \leq x < 36 0^{\circ}$ .
[3 marks]
Answer
A3
Today's identity
Divide every term of $sin^{2} x + cos^{2} x \equiv 1$ by $cos^{2} x$ . What identity do you get?
[2 marks]
Answer
A4
Last lesson
Find the exact value of $cosec (- \frac{2 π}{3}) + cot \frac{π}{4}$ .
[3 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Prove that $(cosec x - sin x) tan x \equiv cos x$ .

Hint Write everything in sin and cos, then use 1 − sin² x = cos² x.

B1
Prove that $(sec x - cos x) cot x \equiv sin x$ .
[3 marks]

Example 2

Prove that $\frac{1 + tan ^{2} x}{1 + cot ^{2} x} \equiv tan^{2} x$ .

Hint Use the two new identities on the top and on the bottom first.

B2
Prove that $sec^{4} x - tan^{4} x \equiv sec^{2} x + tan^{2} x$ .
[3 marks]

Example 3

Solve $2 tan^{2} x + sec x = 1$ for $0^{\circ} \leq x < 36 0^{\circ}$ . Give your answers to 1 decimal place.

Hint Replace tan² x by sec² x − 1 so the equation is a quadratic in sec x.

B3
Solve $2 cot^{2} x - cosec x - 4 = 0$ for $0^{\circ} < x < 36 0^{\circ}$ . Give your answers to 1 decimal place.
[6 marks]
Answer

Example 4

Solve $cot (2 x + \frac{π}{3}) = 1$ for $0 \leq x \leq π$ . Give your answers in terms of $π$ .

Hint Let u = 2x + π/3 and find the interval u lives in before you solve tan u = 1.

B4
Solve $cosec (2 x - \frac{π}{4}) = 2$ for $0 \leq x \leq π$ . Give your answers in terms of $π$ .
[5 marks]
Answer

Example 5

Given that $cosec A = - 5$ and $\frac{3 π}{2} < A < 2 π$ , find the exact values of $cot A$ and $sec A$ .

Hint Use cosec² A = 1 + cot² A, then choose the sign that fits the quadrant.

B5
Given that $sec B = - 10$ and $\frac{π}{2} < B < π$ , find the exact values of $tan B$ and $cosec B$ .
[4 marks]
Answer

Practice

A-Level

C1
Ben solves $tan x sec x = 2 tan x$ for $0 \leq x \leq 2 π$ . He divides both sides by $tan x$ , gets $sec x = 2$ , and gives the answers $x = \frac{π}{3}$ and $x = \frac{5 π}{3}$ . Explain why Ben's answer is not complete, and give all the solutions.
[5 marks]
Answer

C2
Leah solves $tan^{2} x = 2 sec x + 2$ for $0^{\circ} \leq x < 36 0^{\circ}$ . She writes $tan^{2} x = 1 - sec^{2} x$ and finds only $x = 18 0^{\circ}$ . Find Leah's mistake and solve the equation correctly, giving your answers to 1 decimal place.
[5 marks]
Answer
C3
Solve $2 cot^{2} x = 7 cosec x + 2$ for $- 18 0^{\circ} \leq x \leq 18 0^{\circ}$ . Give your answers to 1 decimal place.
[5 marks]
Answer
C4
Find a Cartesian equation for each curve.
(a)
$x = 3 sec t$ , $y = 2 tan t$
[3 marks]
Answer
(b)
$x = 1 + cot t$ , $y = cosec t$
[2 marks]
Answer
Total for C4: 5 marks
C5
Prove that $\frac{1 + sec x}{tan x + sin x} \equiv cosec x$ .
[4 marks]

Exam-style questions

Edexcel exam style

D1
This question is about the equation $tan^{2} θ = 3 sec θ - 3$ .
(a)
Show that the equation can be written as $sec^{2} θ - 3 sec θ + 2 = 0$ .
[2 marks]
(b)
Hence solve $tan^{2} θ = 3 sec θ - 3$ for $0^{\circ} \leq θ < 36 0^{\circ}$ .
[4 marks]
Answer
(c)
Hence solve $tan^{2} (2 x - 3 0^{\circ}) = 3 sec (2 x - 3 0^{\circ}) - 3$ for $0^{\circ} \leq x < 36 0^{\circ}$ .
[4 marks]
Answer
Total for D1: 10 marks

D2
Mia is told that $cot θ = 22$ and $π < θ < \frac{3 π}{2}$ . She writes: " $cosec^{2} θ = 1 + cot^{2} θ = 9$ , so $cosec θ = 3$ ."
Mia is wrong.
(a)
Explain Mia's mistake and give the correct value of cosec θ.
[2 marks]
(b)
Find the exact value of $sec θ$ , giving your answer with a rational denominator.
[3 marks]
Answer
(c)
Show that $cot θ + tan θ \equiv cosec θ sec θ$ . Hence find the exact value of $cot θ + tan θ$ , in the form $\frac{a 2}{b}$ .
[4 marks]
Answer
Total for D2: 9 marks
D3
Solve $cosec^{2} x + cot x = 3$ for $0 \leq x < 2 π$ . Give exact answers where you can, and the others to 3 significant figures.
[7 marks]
Answer
D4
(a)
Prove that $(1 - cos x) (1 + sec x) \equiv sin x tan x$ .
[3 marks]
(b)
Hence solve $(1 - cos x) (1 + sec x) = \frac{3}{2}$ for $0 \leq x \leq 2 π$ , giving your answers in terms of $π$ .
[6 marks]
Answer
Total for D4: 9 marks
D5
A curve C has parametric equations $x = 2 + sec t$ , $y = tan t$ , $- \frac{π}{2} < t < \frac{π}{2}$ .
(a)
Show that a Cartesian equation of C is $(x - 2)^{2} - y^{2} = 1$ .
[3 marks]
(b)
State the range of values that x takes on C, giving a reason.
[2 marks]
Answer
(c)
Hence find the coordinates of the points where C meets the line $y = 2 x - 6$ .
[5 marks]
Answer
Total for D5: 10 marks
D6
(a)
Show that $sec^{2} θ + cosec^{2} θ \equiv sec^{2} θ cosec^{2} θ$ .
[3 marks]
(b)
Hence solve $sec^{2} θ + cosec^{2} θ = \frac{16}{3}$ for $0^{\circ} < θ < 18 0^{\circ}$ .
[5 marks]
Answer
Total for D6: 8 marks

Extension

Stretch

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

E1
Solve $sec x + tan x = 2$ for $0 \leq x < 2 π$ , giving your answer to 3 significant figures.
[8 marks]
Hint 1 · What to try
Write the equation in $sin x$ and $cos x$ and multiply by $cos x$ : $1 + sin x = 2 cos x$ .
Hint 2 · The first line
Square both sides and use $cos^{2} x = 1 - sin^{2} x$ to get a quadratic in $sin x$ .
Hint 3 · The full method
Squaring can add roots. Put each value back into $sec x + tan x = 2$ , and remember $cos x$ cannot be $0$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can derive 1 + tan²x ≡ sec²x and 1 + cot²x ≡ cosec²x from sin²x + cos²x ≡ 1.
I can prove an identity by writing everything in terms of sin x and cos x.
I can solve a quadratic equation in sec, cosec or cot, rejecting values that are impossible.
I can find sec, cosec and cot of an angle from one of them and the quadrant.

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