Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
nta_pyq_2025_jan
Grade 11
Question:
If \alpha > \beta > \gamma > 0, then the expression cot 2 2 2 (1+\beta ) (1+\gamma ) (1+\alpha ) -1 {\beta + (\alpha-\beta) } + cot -1 {\gamma + (\beta-\gamma) }+ cot -1 {\alpha + (\gamma-\alpha) } is equal to :
\pi
0
\pi 2 - (\alpha + \beta + \gamma)
3\pi
Step-by-Step Solution
Key Concept: Apply the core result for inverse trigonometric equations and simplify using the given constraints.
\Rightarrow cot -1 ( \alpha\beta + 1 ) + cot -1 ( \beta\gamma + 1 ) + cot -1 ( \alpha\gamma + 1 ) (1) \alpha - \beta \beta - \gamma \gamma - \alpha \alpha - \beta \beta - \gamma \gamma - \alpha -1 -1 -1 \Rightarrow tan ( ) + tan ( ) + \pi + tan ( ) 1 + \alpha\beta 1 + \beta\gamma 1 + \gamma\alpha -1 -1 -1 -1 -1 -1 \Rightarrow (tan \alpha - tan \beta) + (tan \beta - tan \gamma) + (\pi + tan \gamma - tan \alpha) \Rightarrow \pi
Correct Answer: 1