The value of sin(cot^{-1}(tan(cos^{-1}x))) is | vedclass

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(A) Let $\cos^{-1}x = 	heta$,where $x \in [-1, 1]$.Then $x = \cos 	heta$,which implies $\cos 	heta = x$.We need to evaluate $\sin(\cot^{-1}(	an

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$x$

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(A) Let $\cos^{-1}x = 	heta$,where $x \in [-1, 1]$.Then $x = \cos 	heta$,which implies $\cos 	heta = x$.We need to evaluate $\sin(\cot^{-1}(	an 	heta))$.Since $	an 	heta = \frac{\sin 	heta}{\cos 	heta} = \frac{\sqrt{1 - \cos^2 	heta}}{\cos 	heta} = \frac{\sqrt{1 - x^2}}{x}$.Now,substitute this into the expression:$\sin(\cot^{-1}(\frac{\sqrt{1 - x^2}}{x}))$.Let $\cot^{-1}(\frac{\sqrt{1 - x^2}}{x}) = \phi$.Then $\cot \phi = \frac{\sqrt{1 - x^2}}{x}$.Using the identity $\csc^2 \phi = 1 + \cot^2 \phi = 1 + \frac{1 - x^2}{x^2} = \frac{x^2 + 1 - x^2}{x^2} = \frac{1}{x^2}$.Thus,$\csc \phi = \frac{1}{x}$,which means $\sin \phi = x$.Therefore,the value is $x$.

The value of $\sin(\cot^{-1}(\tan(\cos^{-1}x)))$ is equal to

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