Prove that : ( A + A)/( A - A) + ( A - A)/( A + A) = (2)/(2² A - 1)

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

CBSE Class 10 Mathematics Standard 2025: question 27(b)

Set in the CBSE Class 10 board exam of 2025, paper 30/1/1, section C, question 27(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27(b) · 3 marks · Short answerIntroduction to Trigonometry

Prove that : $\frac{s i n A + c o s A}{s i n A - c o s A} + \frac{s i n A - c o s A}{s i n A + c o s A} = \frac{2}{2 s i n ^{2} A - 1}$

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

Hints

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

Hint 1

Add the two fractions by taking their LCM.

Hint 2

After expanding the squares, the 2 sin A cos A terms cancel. Write cos²A as 1 − sin²A in the denominator.

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.

Combine the two fractions over a common denominator:
LHS $= \frac{( sin A + cos A ) ^{2} + ( sin A - cos A ) ^{2}}{( sin A - cos A ) ( sin A + cos A )}$
1 mark
Expand the numerator and the denominator:
$= \frac{sin ^{2} A + cos ^{2} A + 2 sin A cos A + sin ^{2} A + cos ^{2} A - 2 sin A cos A}{sin ^{2} A - cos ^{2} A}$
1 mark
Use $sin^{2} A + cos^{2} A = 1$ and $cos^{2} A = 1 - sin^{2} A$ :
$= \frac{1 + 1}{sin ^{2} A - ( 1 - sin ^{2} A )}$
½ mark
$= \frac{2}{2 sin ^{2} A - 1} =$ RHS
½ mark

Answer: LHS = 2/(2sin²A − 1) = RHS (proved)

Where the marks are

Takes the LCM correctly (1 mark)
Expands the squares and uses (a − b)(a + b) for the denominator (1 mark)
Applies the identity sin²A + cos²A = 1 (½ mark)
Reaches 2/(2sin²A − 1) (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 30/1/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2025: question 27(b)

Hints

Step-by-step solution

Revise the chapter

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