A) 0
B) \[-1\]
C) \[-2\]
D) 1
Correct Answer: C
Solution :
\[\underset{\alpha \to 0}{\mathop{\lim }}\,\frac{\begin{align} & \cos e{{c}^{-1}}(sec\alpha )+co{{t}^{-1}}(tan\alpha ) \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+co{{t}^{-1}}\cos (si{{n}^{-1}}\alpha ) \\ \end{align}}{\alpha }\] \[=\underset{\alpha \to 0}{\mathop{\lim }}\,\frac{\left[ \begin{align} & \cos e{{c}^{-1}}\left( \cos ec\left( \frac{\pi }{2}-\alpha \right) \right) \\ & +{{\cot }^{-1}}\left( \cot \left( \frac{\pi }{2}-\alpha \right) \right) \\ & +\,{{\cot }^{-1}}\cos [co{{s}^{-1}}\sqrt{1-{{\alpha }^{2}}}] \\ \end{align} \right]}{\alpha }\] \[=\underset{\alpha \to 0}{\mathop{\lim }}\,\frac{\frac{\pi }{2}-\alpha +\frac{\pi }{2}-\alpha +{{\cot }^{-1}}\sqrt{1-{{\alpha }^{2}}}}{\alpha }\] \[=\underset{\alpha \to 0}{\mathop{\lim }}\,\frac{-2-\frac{1}{1+1-{{\alpha }^{2}}}\left( \frac{1}{2\sqrt{1-{{\alpha }^{2}}}}(-2\alpha ) \right)}{1}\] \[=-2\]
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