Question

Rewrite the expression as an algebraic function of x and y

sin(cos^-1(x) - tan^-1(y))

Answer 1

Hmm this is a little tricky...
When we have inverse trig functions, we solve them for angles.
So we can think of each of our inverse trig functions as an angle.

\[\Large\bf\sf \sin\left[\arccos x-\arccos y\right]\quad=\quad \sin\left[\alpha-\beta\right]\]Where,\[\Large\bf\sf \arccos x=\alpha, \qquad\qquad \arctan y= \beta\]

Answer 2

Do you remember the `Sine Angle Addition Formula`?

Answer 3

No

Answer 4

`Angle Addition Formula for Sine`:
\[\Large\bf\sf \color{royalblue}{\sin(a-b)\quad=\quad \sin a \cos b - \sin b \cos a}\]

Answer 5

\[\Large\bf\sf \arccos x=\alpha \qquad\implies\qquad \cos \alpha=x\]We can draw a triangle with this relationship,\[\Large\bf\sf \cos \alpha\quad=\quad \frac{x}{1}\quad=\quad \frac{adjacent}{hypotenuse}\]