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If \(\rm sin (tan^{-1}\ \frac {1}{10} \ + \ cot^{-1} \ x) = 1\) then, find the value of x
1. \(\rm 1\over5\)
2. \(\rm 1\over10\)
3. 10
4. \(\rm 1\over9\)

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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 = \(\rm 1\over10\)

 \(\rm sin^{-1} \ x + cos^{-1} \ x = \frac {π}{2}\)

\(\rm cosec^{-1} \ x +sect^{-1} \ x = \frac {π}{2}\)

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