Prove that: (θ - 2³θ)/(θ - 2³θ) + θ = 0.

MathsCBSE Class 10 · Standard · Board exam 2025, set 30/6/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2025: question 30(a)

Set in the CBSE Class 10 board exam of 2025, paper 30/6/1, section C, question 30(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30(a) · 3 marks · Short answerIntroduction to Trigonometry

Prove that: $\frac{c o s θ - 2 c o s ^{3} θ}{s i n θ - 2 s i n ^{3} θ} + cot θ = 0$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/6/1.

Hints

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

Hint 1

Factor $cos θ$ from the numerator and $sin θ$ from the denominator.

Hint 2

Replace 1 by $sin^{2} θ + cos^{2} θ$ inside the brackets; the bracket becomes $- 1$ .

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 $= \frac{cos θ ( 1 - 2 cos ^{2} θ )}{sin θ ( 1 - 2 sin ^{2} θ )} + cot θ$
½ mark
Writing $1 = sin^{2} θ + cos^{2} θ$ : $= cot θ [\frac{sin ^{2} θ + cos ^{2} θ - 2 cos ^{2} θ}{sin ^{2} θ + cos ^{2} θ - 2 sin ^{2} θ}] + cot θ$
1 mark
$= cot θ \cdot \frac{sin ^{2} θ - cos ^{2} θ}{cos ^{2} θ - sin ^{2} θ} + cot θ$
1 mark
$= - cot θ + cot θ = 0 =$ RHS. Hence proved.
½ mark

Answer: LHS = 0 = RHS (proved)

Where the marks are

Takes $cos θ$ and $sin θ$ common (½ mark)
Replaces 1 by $sin^{2} θ + cos^{2} θ$ (1 mark)
Simplifies to $\frac{s i n ^{2} θ - c o s ^{2} θ}{c o s ^{2} θ - s i n ^{2} θ}$ (1 mark)
Gets $- cot θ + cot θ = 0$ (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 paper2025 board paper, set 30/6/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2025: question 30(a)

Hints

Step-by-step solution

Revise the chapter

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