Trigonometry & Inverse Trigonometry
Trigonometry
star_batch_jee_advanced_2025
Grade 12
Question:
If $[\sin^{-1}\cos x - \sin^{-1}\tan x - 1] = 1$ whose $[.]$ denotes the greatest integer function, then $x$ belongs to:
\tan\sin\cos 1, \tan\sin\cos\sin 1
\tan\cos\sin 1, \tan\cos\sin\cos 1
-1, 1
\sin\cos\tan 1, \sin\cos\sin\tan 1
Step-by-Step Solution
Key Concept: The greatest integer function constrains the range of the composite inverse function, which must lie between 1 and $\frac{\pi}{2}$.
Given $[\sin^{-1} \cos^{-1} \sin^{-1} \tan^{-1} x] = 1$, we establish that $1 \leq \sin^{-1} \cos^{-1} \sin^{-1} \tan^{-1} x \leq \frac{\pi}{2}$. Working backward through the inverse functions with the constraint $\sin \leq \cos^{-1} \sin^{-1} \tan^{-1} x \leq 1$, we find $\sin \cos \sin^{-1} \tan^{-1} x \geq x \tan \sin \cos \sin^{-1}$. The solution requires $x \in [\tan \sin \cos \sin 1]$.
Correct Answer: 1