Prove that (1 + θ - θ)/(1 + θ + θ) = (1 - θ)/(θ).

LHS = θ - θ = (1-θ)/(θ) = RHS

MathsCBSE Class 10 · Standard · Board exam 2024, set 30/2/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2024: question 29

Set in the CBSE Class 10 board exam of 2024, paper 30/2/1, section C, question 29, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29 · 3 marks · Short answerIntroduction to Trigonometry

Prove that $\frac{1 + s e c θ - t a n θ}{1 + s e c θ + t a n θ} = \frac{1 - s i n θ}{c o s θ}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/2/1.

Hints

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

Hint 1

Think of the identity $sec^{2} θ - tan^{2} θ = 1$ .

Hint 2

Replace the 1 in the numerator and take $(sec θ - tan θ)$ common.

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.

Replace $1$ in the numerator by $sec^{2} θ - tan^{2} θ$ : LHS $= \frac{( s e c ^{2} θ - t a n ^{2} θ ) + ( s e c θ - t a n θ )}{1 + s e c θ + t a n θ}$
1 mark
$= \frac{( s e c θ - t a n θ ) ( s e c θ + t a n θ ) + ( s e c θ - t a n θ )}{1 + s e c θ + t a n θ} = \frac{( s e c θ - t a n θ ) ( s e c θ + t a n θ + 1 )}{1 + s e c θ + t a n θ} = sec θ - tan θ$
1 mark
$= \frac{1}{c o s θ} - \frac{s i n θ}{c o s θ} = \frac{1 - s i n θ}{c o s θ} =$ RHS. Hence proved.
1 mark

Answer: LHS = $sec θ - tan θ = \frac{1 - s i n θ}{c o s θ}$ = RHS

Where the marks are

Uses $1 = sec^{2} θ - tan^{2} θ$ in the numerator (1 mark)
Factorises and cancels to get $sec θ - tan θ$ (1 mark)
Converts to $\frac{1 - s i n θ}{c o s θ}$ (1 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 30/2/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2024: question 29

Hints

Step-by-step solution

Revise the chapter

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