The problem is that you have this:

 
but  is false!!!

Edwin

You typed this:

but I don't think you meant to type =-
I think you might have meant this instead:

Put A in the first blank and C in the second:



Use the identity 



Distribute the A



For this to be an identity and true for all x, the term on the left that
contains sin(5x) must equal the term on the right that contains sin(5x). So:



Divide both sides by sin(5x)



Now we'll do it for the other two terms
{
{{A*sin(5x)cos(C)+A*cos(5x)sin(C)}}}

For this to be an identity and true for all x, the term on the left that
contains cos(5x) must equal the term on the right that contains cos(5x). So:



Divide both sides by cos(5x)



Divide equals by equals



Cancel the A's





So C is the angle whose tangent is 2 

So C = arctan(2)

We draw a right triangle that has contains angle C.  Since we know that
tangent = opposite/adjacent, we draw the side opposite C as 2 and the side
adjacent to C as 1.  Then we can use the Pythagorean theorem to find the
hypotenuse:





Since 

Then from the triangle, 





So the answer is:



[Let me know in the space below if this was not what you meant instead of what
 you typed.  If you do, I'll get back to you by email.]

Edwin