Question

Solve for x:
cos(tan^-1x)=sin(cot^-13/4)

Answer 1

Dear student,

$$\begin{gathered} \cos \left({\tan }^{-1}\mathrm{x}\right)=\sin \left[{\cot }^{-1}\left(\frac{3}{4}\right)\right]...\left(1\right) \\ \mathrm{As}\mathrm{we}\mathrm{know}\mathrm{that}\mathrm{sinx}=\cos \left(\frac{\mathrm{\pi }}{2}-\mathrm{x}\right),\mathrm{so} \\ \sin \left[{\cot }^{-1}\left(\frac{3}{4}\right)\right]=\cos \left[\frac{\mathrm{\pi }}{2}-{\cot }^{-1}\left(\frac{3}{4}\right)\right] \\ \mathrm{Now}\mathrm{we}\mathrm{know}\mathrm{that}{\tan }^{-1}\mathrm{x}+{\cot }^{-1}\mathrm{x}=\frac{\mathrm{\pi }}{2}\mathrm{so} \\ \frac{\mathrm{\pi }}{2}-{\cot }^{-1}\left(\frac{3}{4}\right)={\tan }^{-1}\left(\frac{3}{4}\right)\mathrm{putting}\mathrm{this},\mathrm{we}\mathrm{get} \\ \sin \left[{\cot }^{-1}\left(\frac{3}{4}\right)\right]=\cos \left(\frac{\mathrm{\pi }}{2}-{\cot }^{-1}\left(\frac{3}{4}\right)\right)=\cos \left[{\tan }^{-1}\left(\frac{3}{4}\right)\right] \\ \mathrm{Now},\mathrm{putting}\mathrm{this}\mathrm{value}\mathrm{in}\mathrm{eq}\left(1\right),\mathrm{we}\mathrm{get} \\ \cos \left({\tan }^{-1}\mathrm{x}\right)=\cos \left[{\tan }^{-1}\left(\frac{3}{4}\right)\right] \\ \Rightarrow {\tan }^{-1}\mathrm{x}={\tan }^{-1}\left(\frac{3}{4}\right) \\ \Rightarrow \mathrm{x}=\frac{3}{4} \end{gathered}$$

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