Is there a formula for adding say: $A\sin^{-1}C-B\sin^{-1}D$ where A, B, C and D are constants? I'm looking for a formula or a method that can deal with the case when $A\ne B$.
$$A\sin^{-1}C-B\sin^{-1}D$$
There is a formula for when $A=B$:
$$\sin^{-1}(C)-\sin^{-1}(D)=\sin^{-1}(C\sqrt{1-D^2}-D\sqrt{1-C^2})$$
The specific case I have is: $$4\sin^{-1}(38r)-3\sin^{-1}(52r)-\sin^{-1}(-53r)$$
Where $r$ is some constant.

I don’t know about the general case, but for your specific case you can do the following:
First notice that $$ \sin(4x) = 2\sin 2x\cos 2x \\= 4\sin x \cos x (1-2\sin^2 x) \\= 4\sin x(1-2\sin^2x)\sqrt{1-\sin^2 x} $$
So if  $4\sin^{-1} x = y$, then
$$\sin(4\sin^{-1} x) = \sin y = 4x(1-2x^2)\sqrt{1-x^2} \\ \implies y=\sin^{-1} \left(4x(1-2x^2)\sqrt{1-x^2}\right)$$ It can be shown similarly that $$3\sin^{-1}(x)= \sin^{-1} (3x-4x^3) $$
Hence, $$4\sin^{-1}(38r)-3\sin^{-1}(52r)-\sin^{-1}(-53r) \\= \sin^{-1} \left(4(38r)(1-2(38r)^2\sqrt{1-(38r)^2} \right)-\sin^{-1} \left(3(52r)-4(52r)^3\right) +\sin^{-1} (53r) $$ Now you can use the formula you mentioned.
