Prove the following trigonometric identity: 1+cosec Acosec A = (² A)/(1- A)

LHS = 1 + A = (² A)/(1- A) = RHS. Hence proved.

MathsCBSE Class 10 · Standard · Board exam 2025, set 30/5/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2025: question 30(a)

Set in the CBSE Class 10 board exam of 2025, paper 30/5/1, section C, question 30(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30(a) · 3 marks · Short answerIntroduction to Trigonometry

Prove the following trigonometric identity:

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

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/5/1.

Hints

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

Hint 1

Convert cosec A into sin A and simplify the LHS.

Hint 2

The RHS has $1 - sin A$ in the denominator, so multiply and divide by it.

Hint 3

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

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 $= \frac{1 + \frac{1}{s i n A}}{\frac{1}{s i n A}}$ [using $cosec A = \frac{1}{s i n A}$ ]
½ mark
$= \frac{\frac{s i n A + 1}{s i n A}}{\frac{1}{s i n A}} = 1 + sin A$
1 mark
$= \frac{( 1 + s i n A ) ( 1 - s i n A )}{1 - s i n A}$ [multiplying and dividing by $1 - sin A$ ]
½ mark
$= \frac{1 - s i n ^{2} A}{1 - s i n A}$
½ mark
$= \frac{c o s ^{2} A}{1 - s i n A}$ = RHS [since $1 - sin^{2} A = cos^{2} A$ ]. Hence proved.
½ mark

Answer: LHS = $1 + sin A$ = $\frac{c o s ^{2} A}{1 - s i n A}$ = RHS. Hence proved.

Where the marks are

Writes cosec A as $\frac{1}{s i n A}$ (½ mark)
Simplifies LHS to $1 + sin A$ (1 mark)
Multiplies and divides by $1 - sin A$ (½ mark)
Gets $\frac{1 - s i n ^{2} A}{1 - s i n A}$ (½ mark)
Uses the identity to reach RHS (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 30/5/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2025: question 30(a)

Hints

Step-by-step solution

Revise the chapter

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