Question 1024326
.
Write the given expression in terms of x and y only! 

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

PLEASE HELP !!!
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


<pre>
1.  For your information: sin^-1(x) = arcsin(x);   cos^-1(y) = arccos(y).


2.  So, you are given {{{alpha}}} = arcsin(x)  and  {{{beta}}} = arccos(y), 
    and you are asked to express {{{sin(alpha+beta)}}} in terms of x and y.


3.  Since  {{{alpha}}} = arcsin(x)  and  {{{beta}}} = arccos(y),  you have
    {{{sin(alpha)}}} = x  and  {{{cos(beta)}}} = y.

    It implies that  {{{cos(alpha)}}} = +/-{{{sqrt(1-x^2)}}}  (depending on in which quadrant the angle {{{alpha}}} is) 
    and  {{{sin(beta)}}} = +/-{{{sqrt(1-y^2)}}} (depending on where the angle {{{beta}}} is).


4. Now use the formula  {{{sin(alpha + beta)}}} = {{{sin(alpha)*cos(beta) + cos(alpha)*sin(beta)}}}. 

   (If you don't know it, then see the lesson <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson>Addition and subtraction formulas</A> in this site).


   Substitute what you know (what is given and what is written above) into this formula, and you will get 

   {{{sin(alpha + beta)}}} = {{{x*y +- sqrt(1-x^2)*sqrt(1-y^2)}}}.

   with different possible combinations of signs.

   It is your answer.
</pre>