2026 · Q50
If p = sec θ + tan θ and q = cosec θ − cot θ, then what is (p − q − pq) equal to?
Elementary MathematicsTrigonometry
- A−1
- B0
- C1
- D2
Explanation
Test θ = 45°: p = sec 45° + tan 45° = √2 + 1 and q = cosec 45° − cot 45° = √2 − 1, with pq = (√2 + 1)(√2 − 1) = 1. Then p − q − pq = (√2 + 1) − (√2 − 1) − 1 = 1. This identity holds generally (p = (1 + sin θ)/cos θ, q = (1 − cos θ)/sin θ), so the answer is 1, option C.