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

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

CBSE Class 10 Mathematics Basic 2026: question 29

Set in the CBSE Class 10 board exam of 2026, paper 430/3/1, section C, question 29, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29 · 3 marks · Short answerIntroduction to Trigonometry

Prove that: $\frac{1 + cosec A}{cosec A} = \frac{cos ^{2} A}{1 - sin A}$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 430/3/1.

Hints

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

Hint 1

Change cosec A into 1/sin A and simplify the compound fraction.

Hint 2

You need cos²A in the numerator: multiply and divide by (1 − sin 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 $= \frac{1 + \frac{1}{sin A}}{\frac{1}{sin A}}$
½ mark
$= \frac{sin A + 1}{sin A} \times sin A = 1 + sin A$
1 mark
$= \frac{( 1 + sin A ) ( 1 - sin A )}{1 - sin A}$
½ mark
$= \frac{1 - sin ^{2} A}{1 - sin A} = \frac{cos ^{2} A}{1 - sin A}$ = RHS [since $sin^{2} A + cos^{2} A = 1$ ]. Hence proved.
1 mark

Answer: LHS = 1 + sin A = cos²A/(1 − sin A) = RHS (proved)

Where the marks are

Writes cosec A as 1/sin A (½ mark)
Simplifies LHS to 1 + sin A (1 mark)
Multiplies and divides by (1 − sin A) (½ mark)
Uses 1 − sin²A = cos²A to reach the RHS (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 430/3/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2026: question 29

Hints

Step-by-step solution

Revise the chapter

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