Prove that : (θ - cosec θ)² = (1-θ)/(1+θ).

LHS simplifies to (1-θ)/(1+θ) = RHS

MathsCBSE Class 10 · Basic · Board exam 2024, set 430/1/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2024: question 31

Set in the CBSE Class 10 board exam of 2024, paper 430/1/1, section C, question 31, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31 · 3 marks · Short answerIntroduction to Trigonometry

Prove that : $(cot θ - cosec θ)^{2} = \frac{1 - c o s θ}{1 + c o s θ}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/1/1.

Hints

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

Hint 1

Express $cot θ$ and $cosec θ$ using $sin θ$ and $cos θ$ .

Hint 2

Replace $sin^{2} θ$ by $1 - cos^{2} θ$ and factorise it.

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 $= (\frac{cos θ}{sin θ} - \frac{1}{sin θ})^{2}$
1 mark
$= \frac{( cos θ - 1 ) ^{2}}{sin ^{2} θ} = \frac{( 1 - cos θ ) ^{2}}{1 - cos ^{2} θ}$ [since $sin^{2} θ = 1 - cos^{2} θ$ ]
1 mark
$= \frac{( 1 - cos θ ) ^{2}}{( 1 - cos θ ) ( 1 + cos θ )}$
½ mark
$= \frac{1 - cos θ}{1 + cos θ} =$ RHS. Proved.
½ mark

Answer: LHS simplifies to $\frac{1 - c o s θ}{1 + c o s θ}$ = RHS

Where the marks are

Writes cot and cosec in terms of sin and cos (1 mark)
Combines to $\frac{( c o s θ - 1 ) ^{2}}{s i n ^{2} θ}$ (1 mark)
Uses $sin^{2} θ = (1 - cos θ) (1 + cos θ)$ (½ mark)
Cancels to reach RHS (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 430/1/1 (Basic): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Basic 2024: question 31

Hints

Step-by-step solution

Revise the chapter

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