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Question Number 96845 by DELETED last updated on 05/Jun/20

$$\mathrm{If}{\sin }^{-1}\frac{x}{5}+{\mathrm{cosec}}^{-1}\frac{5}{4}=\frac{\pi }{2},\mathrm{then}x=$$

Answered by Sourav mridha last updated on 05/Jun/20

$$\Rightarrow {\sin }^{-1}\left(\frac{\mathit{x}}{5}\right)={\cos }^{-1}\left(\frac{4}{5}\right)={\sin }^{-1}\frac{3}{5}$$$$\Rightarrow \mathit{x}=3$$

Answered by 1549442205 last updated on 05/Jun/20

$$\mathrm{putting}{\sin }^{-1}\frac{\mathrm{x}}{5}=\alpha ,{\mathrm{cosec}}^{-1}\frac{5}{4}=\beta$$$$\Rightarrow \sin \alpha =\frac{\mathrm{x}}{5},\frac{1}{\sin \beta }=\frac{5}{4},\alpha +\beta =\frac{\pi }{2}$$$$\Rightarrow \sin \beta =\frac{4}{5}=\sin (\frac{\pi }{2}-\alpha )=\cos \alpha$$$$\Rightarrow 1={\cos }^2\alpha +{\sin }^2\alpha =\frac{16}{25}+\frac{{\mathrm{x}}^2}{25}$$$$\Rightarrow {\mathrm{x}}^2=9\Rightarrow \mathrm{x}=3$$

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