Prove that : ((1)/( A) - A ) ((1)/( A) - A ) = (1)/( A + A)

Both sides equal A A. Hence proved.

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

CBSE Class 10 Mathematics Standard 2025: question 28(b)

Set in the CBSE Class 10 board exam of 2025, paper 30/3/1, section C, question 28(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(b) · 3 marks · Short answerIntroduction to Trigonometry

Prove that : $(\frac{1}{c o s A} - cos A) (\frac{1}{s i n A} - sin A) = \frac{1}{t a n A + c o t A}$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/3/1.

Hints

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

Hint 1

Simplify each bracket on the left by taking the LCM.

Hint 2

Write tan and cot in terms of sin and cos. Show that both sides reduce to the same expression.

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.

$LHS = (\frac{1 - c o s ^{2} A}{c o s A}) (\frac{1 - s i n ^{2} A}{s i n A})$
1 mark
Use $sin^{2} A + cos^{2} A = 1$ :
$LHS = \frac{s i n ^{2} A}{c o s A} \cdot \frac{c o s ^{2} A}{s i n A}$
½ mark
$= sin A cos A$
½ mark
$RHS = \frac{1}{\frac{s i n A}{c o s A} + \frac{c o s A}{s i n A}} = \frac{s i n A c o s A}{s i n ^{2} A + c o s ^{2} A}$
½ mark
$= \frac{s i n A c o s A}{1} = sin A cos A$ . So LHS = RHS.
½ mark

Answer: Both sides equal $sin A cos A$ . Hence proved.

Where the marks are

Takes LCM in each bracket (1 mark)
Uses the identity to get $\frac{s i n ^{2} A}{c o s A} \cdot \frac{c o s ^{2} A}{s i n A}$ (½ mark)
LHS = $sin A cos A$ (½ mark)
RHS written as $\frac{s i n A c o s A}{s i n ^{2} A + c o s ^{2} A}$ (½ mark)
RHS = $sin A cos A$ , so LHS = 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 paper2025 board paper, set 30/3/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2025: question 28(b)

Hints

Step-by-step solution

Revise the chapter

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