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# How can you rewrite this trigonometric expression as an algebraic expression?

## cos(sin^-1(u)-cos^-1(v)

Mar 9, 2018

$\cos \left({\sin}^{- 1} u - {\cos}^{- 1} v\right) = v \sqrt{1 - {u}^{2}} + u \sqrt{1 - {v}^{2}}$

#### Explanation:

Let ${\sin}^{- 1} u = A$ and ${\cos}^{- 1} v = B$

then $\sin A = u$, which means $\cos A = \sqrt{1 - {u}^{2}}$

and $\cos B = v$, which means $\sin B = \sqrt{1 - {v}^{2}}$

Hence, $\cos \left({\sin}^{- 1} u - {\cos}^{- 1} v\right)$

= $\cos \left(A - B\right)$

= $\cos A \cos B + \sin A \sin B$

= $\sqrt{1 - {u}^{2}} \times v + u \times \sqrt{1 - {v}^{2}}$

= $v \sqrt{1 - {u}^{2}} + u \sqrt{1 - {v}^{2}}$