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STD 12 - 2. inverse trigonometric function — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 2. inverse trigonometric function4 Marks
MCQ
The value of $\sin {\cot ^{ - 1}}\tan {\cos ^{ - 1}}x$ is equal to
  • ✓
    $x$
  • B
    $\frac{x }{2}$
  • C
    $2x$
  • D
    None of these
✓

Answer

Correct option: A.
$x$
a
(a) Let ${\cos ^{ - 1}}x = \theta \,\,\, \Rightarrow \,\,x = \cos \theta \,\, \Rightarrow \,\,\sec \theta = \frac{1}{x}$

$ \Rightarrow \,\,\tan \theta = \sqrt {{{\sec }^2}\theta - 1} = \sqrt {\frac{1}{{{x^2}}} - 1} = \frac{1}{x}\sqrt {1 - {x^2}} $

Now $\sin \,\,{\cot ^{ - 1}}\tan \theta = \sin \,\,{\cot ^{ - 1}}\,\left( {\frac{1}{x}\sqrt {1 - {x^2}} } \right)$

Again, putting $x = \sin \theta $

$\sin \,\,{\cot ^{ - 1}}\left( {\frac{1}{x}\sqrt {1 - {x^2}} } \right) = \sin \,\,{\cot ^{ - 1}}\left( {\frac{{\sqrt {1 - {{\sin }^2}\theta } }}{{\sin \theta }}} \right)$

$ = \sin \,\,{\cot ^{ - 1}}\,(\cot \theta ) = \sin \theta = x$.

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