Assertion (A): For 0 < θ ≤ 90°, cosec θ - θ and cosec θ + θ are reciprocal of each other. Reason (R): ²θ - cosec²θ = 1 (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 · Basic · Board exam 2023, set 430/6/1 · 1 mark · Assertion-reason

CBSE Class 10 Mathematics Basic 2023: question 20

Set in the CBSE Class 10 board exam of 2023, paper 430/6/1, 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): $cot^{2} θ - cosec^{2} θ = 1$

(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 430/6/1.

Hints

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

Hint 1

Multiply the two expressions together.

Hint 2

Compare R with the identity $1 + cot^{2} θ = cosec^{2} θ$ .

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.

$(cosec θ - cot θ) (cosec θ + cot θ) = cosec^{2} θ - cot^{2} θ = 1$ , so the two are reciprocals of each other. A is true.
The correct identity is $cosec^{2} θ - cot^{2} θ = 1$ , so $cot^{2} θ - cosec^{2} θ = - 1$ and R is false. Option (C).
1 mark

Answer: Option (C)

Where the marks are

Shows A is true using $cosec^{2} θ - cot^{2} θ = 1$ , and that R is false (1 mark)

Revise the chapter

Chapter: Introduction to Trigonometry (notes, formulas, where marks are lost and every past question)

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 430/6/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2023: question 20

Hints

Step-by-step solution

Revise the chapter

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