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

Question
Upper level algebra
Express $$\cos(\sin^{-1}x-cos^{-1}y)$$ as an algebraic expression in x and y.

2020-10-28
Step1
Consider the given expression as $$\cos(\sin^{-1}x-cos^{-1}y)$$.
$$Let\ \sin^{-1} x = \alpha\ and\ \cos^{-1} y= \beta,\ then\ x=\sin \alpha\ and\ y=\cos \beta$$.
Now the given expression can be write as $$\cos(\alpha — \beta)$$.
Known formula:
l.$$\cos(\alpha — \beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta$$
2.$$\sin^{2} 0+ \cos^{2} 0=1$$
Step 2
Compute the value of $$\cos \alpha\ and\ \sin \beta$$ as follows.
$$\cos \alpha = \sqrt{1-\sin^{2} \alpha}$$
$$=\sqrt{1-x^{2}}$$
$$\sin \beta = \sqrt{1-\cos^{2} \beta}$$
$$=\sqrt{1-y^{2}}$$
Substitute the values of $$\cos \alpha\ and\ \sin \beta$$ in the formula (1).
$$\cos(\alpha — \beta) =\cos \alpha \cos \beta+\sin \alpha \sin \beta$$
$$=(\sqrt{1-x^{2}})(y)+(x)(\sqrt{1-y^{2}})$$
$$=x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}$$
Thus, the expression $$\cos(\sin^{-1}x-cos^{-1}y)$$ as algebraic in x and y as x $$\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}$$

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