Prove that : √((1 - θ)/(1 + θ)) = θ - θ

Proved: √((1-θ)/(1+θ))=θ-θ

MathsCBSE Class 10 · Standard · Board exam 2026, set 30/2/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Standard 2026: question 24(b)

Set in the CBSE Class 10 board exam of 2026, paper 30/2/1, section B, question 24(b), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q24(b) · 2 marks · Very short answerIntroduction to Trigonometry

Prove that : $\frac{1 - s i n θ}{1 + s i n θ} = sec θ - tan θ$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 30/2/1.

Hints

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

Hint 1

Make the expression under the root a perfect square.

Hint 2

Multiply numerator and denominator by $1 - sin θ$ and use $1 - sin^{2} θ = cos^{2} θ$ .

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 $= \frac{1 - sin θ}{1 + sin θ} \times \frac{1 - sin θ}{1 - sin θ}$ (rationalising)
½ mark
$= \frac{( 1 - sin θ ) ^{2}}{1 - sin ^{2} θ}$
½ mark
$= \frac{( 1 - sin θ ) ^{2}}{cos ^{2} θ}$ [since $1 - sin^{2} θ = cos^{2} θ$ ]
½ mark
$= \frac{1 - sin θ}{cos θ} = \frac{1}{cos θ} - \frac{sin θ}{cos θ} = sec θ - tan θ =$ RHS (θ acute)
½ mark

Answer: Proved: $\frac{1 - s i n θ}{1 + s i n θ} = sec θ - tan θ$

Where the marks are

Multiplies numerator and denominator by $(1 - sin θ)$ (½ mark)
Gets $(1 - sin θ)^{2} / (1 - sin^{2} θ)$ (½ mark)
Uses $1 - sin^{2} θ = cos^{2} θ$ (½ mark)
Takes the square root and splits to $sec θ - tan θ$ (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 paper2026 board paper, set 30/2/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2026: question 24(b)

Hints

Step-by-step solution

Revise the chapter

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