Trigonometry & Inverse Trigonometry
Trig Ratios Functions Identities
nta_abhyas_2025
Grade 12
Question:
Let $y = \sin^{-1}(\sin 8) - \tan^{-1}(\tan 10) + \cos^{-1}(\cos 12) - \sec^{-1}(\sec 9) + \cot^{-1}(\cot 6) - \cos ec^{-1}(\cos ec 7)$. If $y$ simplifies to $ar + b$ then $(a - b) =$
Step-by-Step Solution
Key Concept: Inverse trigonometric functions must be evaluated using principal value ranges, requiring angle reduction modulo appropriate multiples of $\pi$.
We have $y = \sin^{-1}(\sin 8) - \tan^{-1}(\tan 10) + \cos^{-1}(\cos 12)$. This simplifies using inverse function properties: $y = \sec^{-1}(\sec 9) + \cot^{-1}(\cot 6) - \cosec^{-1}(\cosec 7)$. Evaluating each term by applying the principal value ranges and simplifying: $y = (3\pi - 8) - (10 - 3\pi) + (4\pi - 12) - (0 - 2\pi) - (6 - \pi) - (7 - 2\pi) = 13\pi - 40$. Therefore $a = 13$ and $b = -40$.
Correct Answer: 53