Prove that (1 + A)/( A) = (² A)/(1 - A).

LHS simplifies to 1 + A = (² A)/(1 - A) = RHS.

MathsCBSE Class 10 · Standard · Board exam 2023, set 30/6/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2023: question 30

Set in the CBSE Class 10 board exam of 2023, paper 30/6/1, 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 $\frac{1 + s e c A}{s e c A} = \frac{s i n ^{2} A}{1 - c o s A}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/6/1.

Hints

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

Hint 1

Write $sec A$ as $\frac{1}{c o s A}$ and simplify the LHS.

Hint 2

To bring in $sin^{2} A$ , multiply and divide by $1 - cos A$ .

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 $= \frac{1 + s e c A}{s e c A} = \frac{1 + \frac{1}{c o s A}}{\frac{1}{c o s A}} = \frac{c o s A + 1}{c o s A} \times cos A = 1 + cos A$ .
1 mark
Multiply and divide by $(1 - cos A)$ : $= \frac{( 1 + c o s A ) ( 1 - c o s A )}{1 - c o s A} = \frac{1 - c o s ^{2} A}{1 - c o s A}$ .
1 mark
Using $sin^{2} A + cos^{2} A = 1$ : $= \frac{s i n ^{2} A}{1 - c o s A} =$ RHS. Hence proved.
1 mark

Answer: LHS simplifies to $1 + cos A = \frac{s i n ^{2} A}{1 - c o s A}$ = RHS.

Where the marks are

Simplifies LHS to $1 + cos A$ (1 mark)
Multiplies and divides by $1 - cos A$ (1 mark)
Uses $1 - cos^{2} A = sin^{2} A$ to reach RHS (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 paper2023 board paper, set 30/6/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2023: question 30

Hints

Step-by-step solution

Revise the chapter

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