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Question Number 75267 by aliesam last updated on 09/Dec/19

Commented by mind is power last updated on 09/Dec/19

$$\mathrm{let}\mathrm{f}\left(\mathrm{x}\right)=\frac{\sqrt{1+\sin \left(\mathrm{x}\right)}+\sqrt{1-\sin \left(\mathrm{x}\right)}}{\sqrt{1+\sin \left(\mathrm{x}\right)}-\sqrt{1-\sin \left(\mathrm{x}\right)}}$$$$\sin \left(\mathrm{x}\right)=2\sin \left(\frac{\mathrm{x}}{2}\right)\cos \left(\frac{\mathrm{x}}{2}\right)$$$$1\underset{-}{+}\sin \left(\mathrm{x}\right)={\left(\cos \left(\frac{\mathrm{x}}{2}\right)\underset{-}{+}\sin \left(\frac{\mathrm{x}}{2}\right)\right)}^2$$$$\mathrm{since}\mathrm{x}\in (0,\frac{\pi }{4})\Rightarrow \sqrt{1-\sin \left(\mathrm{x}\right)}=\mid \cos \left(\frac{\mathrm{x}}{2}\right)-\sin \left(\frac{\mathrm{x}}{2}\right)\mid =\cos \left(\frac{\mathrm{x}}{2}\right)-\sin \left(\frac{\mathrm{x}}{2}\right)$$$$\Rightarrow \sqrt{1+\sin \left(\mathrm{x}\right)}-\sqrt{1-\sin \left(\mathrm{x}\right)}=2\sin \left(\frac{\mathrm{x}}{2}\right)$$$$\sqrt{1+\sin \left(\mathrm{x}\right)}+\sqrt{1-\sin \left(\mathrm{x}\right)}=(\cos \left(\frac{\mathrm{x}}{2}\right)+\sin \left(\frac{\mathrm{x}}{2}\right)+\cos \left(\frac{\mathrm{x}}{2}\right)-\sin \left(\frac{\mathrm{x}}{2}\right))=2\cos \left(\frac{\mathrm{x}}{2}\right)$$$$\mathrm{f}\left(\mathrm{x}\right)=\frac{2\cos \left(\frac{\mathrm{x}}{2}\right)}{2\sin \left(\frac{\mathrm{x}}{2}\right)}=\cot \left(\frac{\mathrm{x}}{2}\right)$$$${\cot }^{-}\left(\mathrm{f}\left(\mathrm{x}\right)\right)={\cot }^{-}\left(\cot \left(\frac{\mathrm{x}}{2}\right)\right)=\frac{\mathrm{x}}{2}$$

Commented by aliesam last updated on 09/Dec/19

$$godblessyousir$$

Commented by mind is power last updated on 09/Dec/19

$$\mathrm{thanx}\mathrm{sir}$$

Commented by peter frank last updated on 09/Dec/19

$$thankyou$$

Commented by mind is power last updated on 09/Dec/19

$$\mathrm{you}\mathrm{are}\mathrm{Wrlcom}$$

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