# Express cos(sin^{-1}x-cos^{-1}y) as an algebraic expression in x and y.

Express $\mathrm{cos}\left({\mathrm{sin}}^{-1}x-co{s}^{-1}y\right)$ as an algebraic expression in x and y.
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Step1
Consider the given expression as $\mathrm{cos}\left({\mathrm{sin}}^{-1}x-{\mathrm{cos}}^{-1}y\right)$.
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Now the given expression can be write as $\mathrm{cos}\left(\alpha —\beta \right)$.
Known formula:
l.$\mathrm{cos}\left(\alpha —\beta \right)=\mathrm{cos}\alpha \mathrm{cos}\beta +\mathrm{sin}\alpha \mathrm{sin}\beta$
2.${\mathrm{sin}}^{2}0+{\mathrm{cos}}^{2}0=1$
Step 2
Compute the value of as follows.
$\mathrm{cos}\alpha =\sqrt{1-{\mathrm{sin}}^{2}\alpha }$
$=\sqrt{1-{x}^{2}}$
$\mathrm{sin}\beta =\sqrt{1-{\mathrm{cos}}^{2}\beta }$
$=\sqrt{1-{y}^{2}}$
Substitute the values of in the formula (1).
$\mathrm{cos}\left(\alpha —\beta \right)=\mathrm{cos}\alpha \mathrm{cos}\beta +\mathrm{sin}\alpha \mathrm{sin}\beta$
$=\left(\sqrt{1-{x}^{2}}\right)\left(y\right)+\left(x\right)\left(\sqrt{1-{y}^{2}}\right)$
$=x\sqrt{1-{y}^{2}}+y\sqrt{1-{x}^{2}}$
Thus, the expression $\mathrm{cos}\left({\mathrm{sin}}^{-1}x-{\mathrm{cos}}^{-1}y\right)$ as algebraic in x and y as x $\sqrt{1-{y}^{2}}+y\sqrt{1-{x}^{2}}$