The value of

Question:

The value of $\tan \left[\cos ^{-1}\left\{\sin \left(\cot ^{-1} 1\right)\right\}\right]$ is ___________________

Solution:

$\tan \left\{\cos ^{-1}\left[\sin \left(\cot ^{-1} 1\right)\right]\right\}$

$=\tan \left\{\cos ^{-1}\left[\sin \left(\frac{\pi}{4}\right)\right]\right\}$                      $\left(\cot \frac{\pi}{4}=1 \Rightarrow \cot ^{-1} 1=\frac{\pi}{4}\right)$

$=\tan \left\{\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)\right\}$

$=\tan \frac{\pi}{4}$                                                                  $\left(\cos \frac{\pi}{4}=\frac{1}{\sqrt{2}} \Rightarrow \cos ^{-1} \frac{1}{\sqrt{2}}=\frac{\pi}{4}\right)$

$=1$

Thus, the value of $\tan \left\{\cos ^{-1}\left[\sin \left(\cot ^{-1} 1\right)\right]\right\}$ is 1 .

The value of $\tan \left[\cos ^{-1}\left\{\sin \left(\cot ^{-1} 1\right)\right\}\right]$ is __1__.

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