$\lim_{x \to \frac{1}{\sqrt{2}}} (\frac{\sin (\cos^{- 1} x) - x}{1 - \tan (\cos^{- 1} x)})$ is equal to

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$\frac{1}{\sqrt{2}}$

$- \frac{1}{\sqrt{2}}$

$- \sqrt{2}$

$\sqrt{2}$

answer is D.

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Detailed Solution

$x = \cos θ, x \to \frac{1}{\sqrt{2}}$ then $θ \to \frac{π}{4}$ then

$\lim_{θ \to \frac{π}{4}} \frac{\sin θ - \cos θ}{1 - \tan θ} = \lim_{θ \to \frac{π}{4}} \frac{(\sin θ - \cos θ) \cos θ}{(\sin θ - \cos θ)} = \lim_{θ \to \frac{π}{4}} (- \cos θ) = - \frac{1}{\sqrt{2}}$

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