2018 · Q57
Consider the following statements:
- 1(sec²θ - 1)(1 - cosec²θ) = 1
- 2sinθ(1 + cosθ)⁻¹ + (1 + cosθ)(sinθ)⁻¹ = 2cosecθ
Which of the above is/are correct?
Elementary MathematicsTrigonometry
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
Statement 1: (sec²θ − 1)(1 − cosec²θ) = tan²θ·(−cot²θ) = −1 ≠ 1, so it is false. Statement 2: combining over sinθ(1 + cosθ) gives [sin²θ + (1 + cosθ)²]/[sinθ(1 + cosθ)] = 2(1 + cosθ)/[sinθ(1 + cosθ)] = 2cosecθ, true. Only statement 2 is correct, option (B).