Q1. [1] § 8.2 Trigonometric Ratios § 8.4 Trigonometric Identities § 8.5 Summary
A statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.
Statement A (Assertion): For $0 < \theta \leq 90°$, $\csc\theta - \cot\theta$ and $\csc\theta + \cot\theta$ are reciprocal of each other.
Statement R (Reason): $\csc^2\theta - \cot^2\theta = 1$
- 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.
Previously asked in CBSE board exam
2023 30/6/1 Q20
Generated by claude-sonnet-4-6 · 2026-06-15 10:27 · grounding rag
Model Answer
Option A
Since $\csc^2\theta - \cot^2\theta = 1$, we get $(\csc\theta - \cot\theta)(\csc\theta + \cot\theta) = 1$, so the two expressions are indeed reciprocals of each other. Both A and R are true, and R directly explains A.
Explanation
The identity $\csc^2\theta - \cot^2\theta = 1$ (from the source, §8.4) can be factorised as a difference of squares to give $(\csc\theta - \cot\theta)(\csc\theta + \cot\theta) = 1$, which is precisely what "reciprocal of each other" means. So R is the correct and complete explanation of A — choose Option A.
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