<s>[INST] Rewrite $\sin(\cos^{-1}(x)-\tan^{-1}(y))$ as an algebraic function of $x$ and $y$. Rewrite the expression as an algebraic function of $x$ and $y$:
$$\sin(\cos^{-1}(x)-\tan^{-1}(y)).$$
I am unsure of how to change this into an algebraic function, yet I am able to simplify inso sin and cos.
thank you
 [/INST] Let $\theta = \arccos x$.  Then $\cos\theta = x$, so we can draw a right triangle with adjacent side of length $|x|$ and hypotenuse of length $1$.  By the Pythagorean Theorem, the opposite side has length $\sqrt{1 - x^2}$.   
Let $\varphi = \arctan y$.  Then $\tan\varphi = y$, so we can draw a right triangle with opposite side of length $|y|$ and adjacent side of length $1$.  By the Pythagorean Theorem, the hypotenuse has length $\sqrt{1 + y^2}$.  
Thus,
\begin{align*}
\sin(\arccos x + \arctan y) & = \sin(\theta + \varphi)\\
                            & = \sin\theta\cos\varphi + \cos\theta\sin\varphi
\end{align*}
From the diagrams, we see that 
\begin{align*}
\cos\theta & = x\\
\sin\theta & = \sqrt{1 - x^2}\\
\cos\varphi & = \frac{1}{\sqrt{1 + y^2}}\\
\sin\varphi & = \frac{y}{\sqrt{1 + y^2}}
\end{align*}
Make the appropriate substitutions, then simplify.
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