Assertion (A): For 0 < θ ≤ 90°, cosec θ - θ and cosec θ + θ are reciprocal of each other. Reason (R): cosec²θ - ²θ = 1 Choose the correct option out of the following: (A) Both Assertion (A) and Reason (R) are true; and Reason (R) is the correct explanation of Assertion (A). (B) Both Assertion (A) and Reason (R) are true; but Reason (R) is not the correct explanation of Assertion (A). (C) Assertion (A) is true but Reason (R) is false. (D) Assertion (A) is false but Reason (R) is true.

MathsCBSE Class 10 · Standard · Board exam 2023, set 30/6/1 · 1 mark · Assertion-reason

CBSE Class 10 Mathematics Standard 2023: question 20

Set in the CBSE Class 10 board exam of 2023, paper 30/6/1, section A, question 20, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q20 · 1 mark · Assertion-reasonCompetency-basedIntroduction to Trigonometry

Assertion (A): For $0 < θ \leq 9 0^{\circ}$ , $cosec θ - cot θ$ and $cosec θ + cot θ$ are reciprocal of each other.

Reason (R): $cosec^{2} θ - cot^{2} θ = 1$

Choose the correct option out of the following:

(A)Both Assertion (A) and Reason (R) are true; and Reason (R) is the correct explanation of Assertion (A).
(B)Both Assertion (A) and Reason (R) are true; but Reason (R) is not the correct explanation of Assertion (A).
(C)Assertion (A) is true but Reason (R) is false.
(D)Assertion (A) is false but Reason (R) is true.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/6/1.

Hints

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

Hint 1

Two numbers are reciprocals if their product is 1.

Hint 2

Factorise $cosec^{2} θ - cot^{2} θ$ as a difference of squares.

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.

R is the standard identity $cosec^{2} θ - cot^{2} θ = 1$ , so R is true.
Factorising R: $(cosec θ - cot θ) (cosec θ + cot θ) = 1$ , so the two expressions are reciprocals. A follows from R. Option (A).
1 mark

Answer: Option (A)

Where the marks are

Factorises the identity to show the product is 1 (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 paper2023 board paper, set 30/6/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2023: question 20

Hints

Step-by-step solution

Revise the chapter

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