Prove that (cosec A – A)( A – A) = (1)/( A + A)

Both sides equal A A; identity proved.

MathsCBSE Class 10 · Basic · Board exam 2026, set 430/4/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2026: question 28(b)

Set in the CBSE Class 10 board exam of 2026, paper 430/4/1, 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 $(cosec A - sin A) (sec A - cos A) = \frac{1}{t a n A + c o t A}$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 430/4/1.

Hints

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

Hint 1

Write cosec A as $\frac{1}{s i n A}$ and sec A as $\frac{1}{c o s A}$ .

Hint 2

Use $1 - sin^{2} A = cos^{2} A$ and $1 - cos^{2} A = sin^{2} A$ .

Hint 3

Write $1$ as $sin^{2} A + cos^{2} A$ in the denominator and divide by $sin A cos A$ .

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 $= (\frac{1}{s i n A} - sin A) (\frac{1}{c o s A} - cos A) = \frac{1 - s i n ^{2} A}{s i n A} \times \frac{1 - c o s ^{2} A}{c o s A}$
1 mark
$= \frac{c o s ^{2} A}{s i n A} \times \frac{s i n ^{2} A}{c o s A} = sin A cos A$ [using $sin^{2} A + cos^{2} A = 1$ ]
1 mark
$= \frac{s i n A c o s A}{s i n ^{2} A + c o s ^{2} A}$ ; dividing numerator and denominator by $sin A cos A$ : $= \frac{1}{\frac{s i n A}{c o s A} + \frac{c o s A}{s i n A}} = \frac{1}{t a n A + c o t A} =$ RHS. Hence proved.
1 mark

Answer: Both sides equal $sin A cos A$ ; identity proved.

Where the marks are

Writes cosec A and sec A in terms of sin and cos and simplifies brackets (1 mark)
Reduces LHS to $sin A cos A$ (1 mark)
Shows $sin A cos A = \frac{1}{t a n A + c o t A}$ (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 paper2026 board paper, set 430/4/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2026: question 28(b)

Hints

Step-by-step solution

Revise the chapter

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