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

MathsCBSE Class 10 · Standard · Sample paper, session 2025-26 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard sample paper 2025-26: question 29(B)

From the official CBSE sample question paper for 2025-26, section C, question 29(B), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29(B) · 3 marks · Short answerIntroduction to Trigonometry

Prove that $\frac{c o s A - s i n A + 1}{c o s A + s i n A - 1} = cosec A + cot A$

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

Look for a factor that turns the denominator into a difference of squares.

Hint 2

Multiply above and below by $(cos A + sin A + 1)$ .

Hint 3

Replace $1 - sin^{2} A$ by $cos^{2} A$ and take out the common factor.

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.

Multiply the numerator and the denominator by $(cos A + sin A + 1)$ :
LHS $= \frac{( cos A + 1 - sin A ) ( cos A + 1 + sin A )}{( cos A + sin A - 1 ) ( cos A + sin A + 1 )}$ .
1 mark
$= \frac{( cos A + 1 ) ^{2} - sin ^{2} A}{( cos A + sin A ) ^{2} - 1} = \frac{cos ^{2} A + 2 cos A + 1 - sin ^{2} A}{2 sin A cos A}$ [using $sin^{2} A + cos^{2} A = 1$ in the denominator].
1 mark
In the numerator, $1 - sin^{2} A = cos^{2} A$ , so it becomes $2 cos^{2} A + 2 cos A = 2 cos A (1 + cos A)$ .
LHS $= \frac{2 cos A ( 1 + cos A )}{2 sin A cos A} = \frac{1 + cos A}{sin A} = \frac{1}{sin A} + \frac{cos A}{sin A} = cosec A + cot A$ = RHS. Hence proved.
1 mark

Answer: LHS = cosec A + cot A (proved)

Where the marks are

Multiplies the numerator and denominator by $(cos A + sin A + 1)$ (1 mark)
Simplifies to $\frac{c o s ^{2} A + 2 c o s A + 1 - s i n ^{2} A}{2 s i n A c o s A}$ (1 mark)
Reduces this to $\frac{1 + c o s A}{s i n A} = cosec A + cot A$ (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 paperSample paper 2025-26 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2025-26: question 29(B)

Hints

Step-by-step solution

Revise the chapter

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