Correct Answer - Option 2 :
\(\rm 1\over10\)
Concept:
sin x = y then x = sin-1 y
\(\rm tan^{-1} \ x + cot^{-1} \ x = \frac {π}{2}\)
Calculation:
Given: \(\rm sin (tan^{-1}\ \frac {1}{10} \ + \ cot^{-1} \ x) = 1\)
⇒ \(\rm tan^{-1}\ \frac {1}{10} \ + \ cot^{-1} \ x = sin^{-1}\ (1)\) (∵ sin-1 (1) = sin-1 (sin (π/2)) = π/2)
⇒ \(\rm tan^{-1}\ \frac {1}{10} \ + \ cot^{-1} \ x = {\pi\over2}\)
Here, \(\rm tan^{-1} \ x + cot^{-1} \ x = \frac {π}{2}\)
Then x = \(\rm 1\over10\)
\(\rm sin^{-1} \ x + cos^{-1} \ x = \frac {π}{2}\)
\(\rm cosec^{-1} \ x +sect^{-1} \ x = \frac {π}{2}\)