CIE AS Maths: Pure 2

Topic Questions

3.4 Trigonometric Proof

1
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2 marks

Show that

cot space thetaidentical to fraction numerator cos space theta over denominator sin space theta end fraction

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2a
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2 marks

Use the identity

cosopen parentheses A plus B close parentheses identical tocos Acos B minussin Asin B

to show that

cos 2 A identical tocos2 A minussin2 A

2b
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3 marks

Show by counter-example that

cos 2 theta not identical tocos theta pluscos theta

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3
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2 marks

Prove the identity

fraction numerator sin space 2 theta over denominator 2 space sin space theta end fraction identical to cos space theta comma space space space space space space space space space space space space theta not equal to kπ

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4
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4 marks

Show that

sin squared space theta open parentheses sec squared space theta plus cosec squared space theta close parentheses identical to sec squared space theta

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5
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5 marks
(i)
Use the quotient rule to show that

fraction numerator d over denominator d x end fraction open square brackets cosec space x close square brackets equals fraction numerator negative cos space x over denominator sin squared space x end fraction

(ii)
Hence show that

fraction numerator d over denominator d x end fraction open square brackets cosec space x close square brackets equals negative cot space x space cosec space x

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6
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3 marks

Show that

3 sin 2 theta minus 2 sin theta identical to 2 sin theta space open parentheses 3 space cos space theta minus 1 close parentheses

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7
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5 marks

Prove the identity

2 cosec 2 xcot x identical tocosec2 x comma space space space space space space space space space space space x not equal to fraction numerator k straight pi over denominator 2 end fraction

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8
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3 marks

Use the identity

R sinopen parentheses theta plus straight alpha close parentheses identical to R spacecos alpha sin theta plus R spacesin alpha cos theta

to show that

4 sinopen parentheses theta plus straight pi over 4 close parentheses identical to 2 square root of 2 open parentheses sin space theta plus cos space theta close parentheses

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1
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4 marks

Given the identity

cosopen parentheses A plus B close parentheses equalscos A cos Bminussin A sin B

prove the following identities:

(i)

cos 2 theta identical tocos2 theta minussin2 theta

(ii)

cos 2 theta identical to 1 minus 2 spacesin2 theta

(iii)
cos 2 theta identical to 2 spacecos2 theta minus 1

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2
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5 marks
(i)
Prove the identity

sin 3 theta identical to 3 sin theta minus 4 sin3 theta

(ii)
Show by counter-example that

cos 3 theta not identical to 3 cos theta minus4 cos3 theta

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3
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5 marks

Show that

cos 4 theta pluscos straight pi over 3 identical to 8 sin4 theta minus 8 sin2 theta plus 3 over 2

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4
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5 marks

Prove that

cot2 theta minustan2 theta identical to 4 cot 2 theta cosec 2 theta.

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5
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5 marks

Prove the identity

fraction numerator 1 minus tan squared space x over denominator cos space 2 x end fraction identical to sec squared space x space space space space space space space space space space space space x not equal to fraction numerator 2 straight k plus 1 over denominator 4 end fraction straight pi

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6
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4 marks

Prove the identity

cosec x identical to fraction numerator 1 half space sec squared x over 2 over denominator tan x over 2 end fraction

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7
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5 marks

Show that

tanx over 2 identical to fraction numerator 1 over denominator cosec space x plus cot space x end fraction space space space space space space space space space space space space space x not equal to 2 kπ

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1
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4 marks

Given the identity

sin2 theta pluscos2 theta identical to 1

prove the following identities:

(i)

sec2 theta identical to 1 plustan2 theta

(ii)
cosec2 theta identical to 1 pluscot2 theta

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2
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5 marks
(i)
By using the double angle formula for cosine, prove the identity

cos 4 theta identical to 8 cos4 theta minus 8 cos2 theta plus 1

(ii)
Show by counter-example that

sin 4 theta not identical to 8 sin4 theta minus 8 sin2 theta plus 1

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3
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3 marks

Prove the identity

fraction numerator 4 space sin to the power of 4 space theta over denominator sin squared space 2 theta end fraction identical to tan squared space theta space space space space space space space space space space theta not equal to kπ

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4
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4 marks

Show that

sin theta open parentheses cosec squared space theta minus 2 close parentheses identical to fraction numerator cos space 2 theta over denominator sin space theta end fraction

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5
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5 marks

Show that

sin 3 theta plussin theta identical to 4 sin theta minus 4 spacesin3 theta

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6
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5 marks

Prove the identity

fraction numerator 4 space cot space x space cos space 2 x over denominator sin space 4 x end fraction identical to cosec squared space x italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space x not equal to kπ over 4

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7
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3 marks

Show that

square root of 2 sin open parentheses theta minus straight pi over 4 close parentheses identical to sin space theta minus cos space theta

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1
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6 marks

Consider the three triangles, all of height 1, as shown below.

q1-5-8-trignometric-proof-a-level-only-edexcel-a-level-pure-maths-veryhard

By applying the area of a triangle formula A equals 1 half ab space sin space C  to each one, prove that,

sinopen parentheses A plus B close parentheses identical tosin A cos B plussin B cos A

Briefly explain why this only proves the result for A and B being acute angles.

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2
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4 marks

Prove the identity

tan space 4 theta identical to fraction numerator 4 space tan space theta open parentheses 1 minus tan squared space theta close parentheses over denominator 1 minus 6 space tan squared space theta plus tan to the power of 4 space theta end fraction

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3
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5 marks

Prove the identity

negative 16 cot 2 theta cosec3 2 theta identical tosec4 theta minuscosec4 theta

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4
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4 marks

Show that

fraction numerator square root of 2 cos open parentheses theta plus straight pi over 4 close parentheses over denominator sin open parentheses theta minus straight pi over 2 close parentheses end fraction identical to tan space theta minus 1

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5a
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4 marks

Show that

sin 3 theta identical to 3 sin straight theta cos2 theta minussin3 theta

5b
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5 marks

Hence, or otherwise, show that

fraction numerator cos space 3 theta minus cos space theta over denominator sin space 3 theta space sin space theta end fraction identical to fraction numerator 4 space cos space theta over denominator 1 minus 4 space cos squared space theta end fraction space space space space space space space space space space theta not equal to kπ

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6
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5 marks

Show that

4cos2 open parentheses x minus straight pi over 6 close parentheses identical to 3 minus 2sin2 x plus square root of 3 sin 2 x

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7
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6 marks

Show that

tanopen parentheses fraction numerator 2 x plus straight pi over denominator 4 end fraction close parentheses identical tosec x plustan x

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8
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9 marks

Show that

1 over open parentheses begin display style fraction numerator square root of 3 over denominator 2 end fraction end style cos space theta minus begin display style 1 half end style sin space theta close parentheses squared plus 1 over open parentheses begin display style fraction numerator square root of 3 over denominator 2 end fraction end style sin space theta plus begin display style 1 half end style cos space theta close parentheses squared identical to 4 space cosec squared open parentheses 2 theta plus straight pi over 3 close parentheses

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