Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
nta_pyq_2025_jan
Grade 11
Question:
If for some \alpha, \beta; \alpha \le \beta, \alpha + \beta - 8 and sec (tan 2 -1 2 \alpha) + cosec (cot -1 \beta) - 36 , then \alpha + \beta is_______. 2
Step-by-Step Solution
Key Concept: Apply the core result for inverse trigonometric equations and simplify using the given constraints.
-1 Let tan \alpha = A \Rightarrow tan A = \alpha -1 (14) cot \beta = B \Rightarrow cot B = \beta 2 2 sec A + cosec B = 36 2 2 \Rightarrow 1 + tan A + 1 + cot B = 36 2 2 \Rightarrow \alpha + \beta = 34 Also \alpha + \beta = 8 (Given) 2 \therefore (\alpha + \beta) = 34 + 2\alpha\beta = 64 \Rightarrow \alpha\beta = 15 \Rightarrow \alpha, \beta are roots of equation 2 x - 8x + 15 = 0 \Rightarrow (x - 3)(x - 5) = 0 \Rightarrow x = 3, 5 \therefore \alpha = 3, \beta = 5 (\alpha < \beta) 2 \therefore \alpha + \beta = 9 + 5 = 14
Correct Answer: 14