SOLUTION: sin^-1 (3÷5) + sin^-1 (5÷13) = sin^-1 (56÷65)

Thus we need to prove that  arcsin + arcsin = arcsin.

Proof

Let  = arcsin and  = arcsin.

Thus  =  and  is in Quadrant 1;    =  and  is in Quadrant 1 also.

We need to prove that  =   and   +  is in Quadrant 1.

Since  = , you have  =  =         (make all intermediate calculations yourself)

Since  = , you have  =  =     (make all intermediate calculations yourself)

Now apply the formula for :

 =  =  =  = .   

Thus the first half of the statement is proved.

Next calculate :

 =  =  =  = .

Since  > 0,  is in the quadrant 1.

The proof is completed.