Prove that : ( A)/(1 + A) - ( A)/(1 - A) = 2 cosec A

LHS simplifies to (2)/( A) = 2 cosec A (proved)

MathsCBSE Class 10 · Standard · Board exam 2026, set 30/1/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2026: question 28(B)

Set in the CBSE Class 10 board exam of 2026, paper 30/1/1, 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 : $\frac{t a n A}{1 + s e c A} - \frac{t a n A}{1 - s e c A} = 2 cosec A$

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

Hints

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

Hint 1

Change tan A to sin A/cos A and sec A to 1/cos A.

Hint 2

After simplifying, take sin A common and combine the two fractions with an LCM.

Hint 3

The denominator cos²A − 1 can be written using sin²A.

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.

Write everything in terms of sin A and cos A:
$LHS = \frac{tan A}{1 + sec A} - \frac{tan A}{1 - sec A} = \frac{\frac{s i n A}{c o s A}}{1 + \frac{1}{c o s A}} - \frac{\frac{s i n A}{c o s A}}{1 - \frac{1}{c o s A}}$
1 mark
Simplify each fraction by multiplying its numerator and denominator by $cos A$ :
$= \frac{sin A}{cos A + 1} - \frac{sin A}{cos A - 1}$
½ mark
Take the LCM:
$= sin A [\frac{( cos A - 1 ) - ( cos A + 1 )}{cos ^{2} A - 1}] = sin A (\frac{- 2}{- sin ^{2} A})$
[since $cos^{2} A - 1 = - sin^{2} A$ ]
1 mark
$= \frac{2}{sin A} = 2 cosec A = RHS$
Hence proved.
½ mark

Answer: LHS simplifies to $\frac{2}{s i n A} = 2 cosec A$ (proved)

Where the marks are

Converts tan A and sec A into sin A and cos A (1 mark)
Simplifies to $\frac{s i n A}{c o s A + 1} - \frac{s i n A}{c o s A - 1}$ (½ mark)
Combines the fractions and uses $cos^{2} A - 1 = - sin^{2} A$ (1 mark)
Reaches $\frac{2}{s i n A} = 2 cosec A$ (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 30/1/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2026: question 28(B)

Hints

Step-by-step solution

Revise the chapter

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