Question

Write the given expression in terms of x and y only.
sin(sin^−1 (x) + cos^−1 (y))

Answer 1

Using the addition formula I get:

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

But I'm not really sure how to go from there.

Answer 2

hmm, i think it might be a little mixed up. The sum formula is:\[\sin(a+b)=\sin a \cos b+ \sin b \cos a\]substituting:\[a=\sin ^{-1} x,b=\cos^{-1}y\]we get:\[\sin (\sin ^{-1}x)\cos(\cos^{-1}y)+\sin(\cos ^{-1}y)+\cos(\sin ^{-1}x)\]

Answer 3

Right. I did switch it up, but how am I supposed to get things in terms of X and Y from there
?

Answer 4

the two terms on the left simplify nicely:\[\sin (\sin ^{-1}x)=x, \cos( \cos^{-1}y)=y\]

Answer 5

What about the second half of the problem.

Answer 6

The ones on the right might require some right triangle drawing.

Answer 7

Continuing after @joemath314159:

\[ \sin (\sin ^{-1}x)\cos(\cos^{-1}y)+\sin(\cos ^{-1}y)+\cos(\sin ^{-1}x) \]

\[\sin (\sin ^{-1}x) = x\]

\[ \cos(\cos^{-1}y) = y\]

\[ \sin(\cos ^{-1}y) = \sqrt{1-y^2} \]

\[ \cos(\sin ^{-1}x = \sqrt{1-x^2} \]

Now substitute.

Answer 8

Ahhh, many thanks.