cos−1x=tan−1√1−x2x, then:

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c o s^{- 1} x = t a n^{- 1} \frac{\sqrt{1 - x^{2}}}{x}

, then:

Options:

A . X∈R

B . −≤x≤1,x≠0

C . 0

D . None of these

Answer: Option C
:
C
Putting

θ = c o s^{- 1} x

in R.H.S., we have
R.H.S

= t a n^{- 1} \frac{\sqrt{1 - x^{2}}}{x} = t a n^{- 1} \frac{\sqrt{1 - c o s^{2} θ}}{c o s θ} (0 \leq θ \leq π) = t a n^{- 1} \frac{s i n θ}{c o s θ} = t a n^{- 1} t a n θ = θ = c o s^{- 1} x

when

- \frac{π}{2} < θ < \frac{π}{2}

i.e., when

0 \leq θ < \frac{π}{2}

i.e., when

0 < c o s θ \leq 1

i.e.,

0 < x \leq 1

.

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