Prove that (θ - 2³θ)/(2³θ - θ) = θ.

MathsCBSE Class 10 · Standard · Board exam 2024, set 30/3/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2024: question 30

Set in the CBSE Class 10 board exam of 2024, paper 30/3/1, section C, question 30, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30 · 3 marks · Short answerIntroduction to Trigonometry

Prove that $\frac{s i n θ - 2 s i n ^{3} θ}{2 c o s ^{3} θ - c o s θ} = tan θ$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/3/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Take $sin θ$ common in the numerator and $cos θ$ common in the denominator.

Hint 2

Rewrite $2 cos^{2} θ - 1$ using $cos^{2} θ = 1 - sin^{2} θ$ and compare it with the numerator's bracket.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

LHS $= \frac{sin θ - 2 sin ^{3} θ}{2 cos ^{3} θ - cos θ} = \frac{sin θ ( 1 - 2 sin ^{2} θ )}{cos θ ( 2 cos ^{2} θ - 1 )}$ [taking common factors]
½ mark
$= \frac{tan θ ( 1 - 2 sin ^{2} θ )}{2 ( 1 - sin ^{2} θ ) - 1}$ [using $\frac{s i n θ}{c o s θ} = tan θ$ and $cos^{2} θ = 1 - sin^{2} θ$ ]
1 mark
$= \frac{tan θ ( 1 - 2 sin ^{2} θ )}{1 - 2 sin ^{2} θ}$
1 mark
$= tan θ =$ RHS. Hence proved.
½ mark

Answer: LHS simplifies to tan θ = RHS (proved)

Where the marks are

Takes out $sin θ$ and $cos θ$ as common factors (½ mark)
Uses $cos^{2} θ = 1 - sin^{2} θ$ and $tan θ = sin θ / cos θ$ (1 mark)
Simplifies the denominator to $1 - 2 sin^{2} θ$ (1 mark)
Cancels and concludes $= tan θ$ (½ mark)

Revise the chapter

Chapter: Introduction to Trigonometry (notes, formulas, where marks are lost and every past question)
Topic: Proving Simple Trigonometric Identities

Practise this chapterBoard-style questions from Introduction to Trigonometry, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 30/3/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2024: question 30

Hints

Step-by-step solution

Revise the chapter

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