Solve the following equation: ( ^ -1 x) = ( ^ -1 3 4 ). - Vidyadip

Question

Solve the following equation:$\cos(\tan^{-1}\text{x}) = \sin\bigg(\cot^{-1}\frac{3}{4}\bigg).$

✓

Answer

Given $\cos(\tan^{-1}\text{x}) = \sin\bigg(\cot^{-1}\frac{3}{4}.\bigg)$
$\Rightarrow\cos(\tan^{-1}\text{x}) = \cos\bigg(\frac{\pi}{2} - \cot^{-1}\frac{3}{4}\bigg)$
$\Rightarrow\tan^{-1}\text{x} = \frac{\pi}{2} - \cot^{-1}\frac{3}{4}$
$\Rightarrow\frac{\pi}{2} - \cot^{-1}\text{x} = \frac{\pi}{2} - \cot^{-1}\frac{3}{4}\Rightarrow\cot^{-1}\text{x} = \cot^{-1}\frac{3}{4}$
$\Rightarrow\text{x} = \frac{3}{4}$
$ \begin{bmatrix} \text{ Note}: \sin\theta = \cos\bigg(\frac{\pi}{2} -\theta\bigg)\\ \tan^{-1}\text{x} + \cot^{-1}\text{x} = \frac{\pi}{2} \end{bmatrix} .$

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