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

Question
Upper level algebra
asked 2020-10-27
Express \(\cos(\sin^{-1}x-cos^{-1}y)\) as an algebraic expression in x and y.

Answers (1)

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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