Question 515399
To find {{{sin(alpha + beta)}}}, we need to use the sum formula for sine:

{{{sin(alpha + beta) = sin(alpha)*cos(beta) + cos(alpha)*sin(beta)}}}

We have {{{cos(alpha) = 4/5}}} and {{{cos(beta) = 12/13}}}, and we know that {{{alpha}}} and {{{beta}}} are first quadrant angles, which implies that both {{{sin(alpha)}}} and {{{sin(beta)}}} are positive.  To find {{{sin(alpha)}}}, recall the trigonometric identity:

{{{(sin(alpha))^2 + (cos(alpha))^2 = 1}}}

Using what we know, we can solve for {{{sin(alpha)}}}:

{{{(sin(alpha))^2 + (4/5)^2 = 1}}} (substituting {{{4/5}}} for {{{cos(alpha)}}})
{{{(sin(alpha))^2 + 16/25 = 1}}} (simplifying)
{{{(sin(alpha))^2 = 1 - 16/25}}} (subtracting {{{16/25}}} from both sides)
{{{(sin(alpha))^2 = 25/25 - 16/25}}} (putting all terms on the right side under a common denominator)
{{{(sin(alpha))^2 = 9/25}}} (simplifying)
{{{sin(alpha) = 3/5}}} (taking the positive square root of both sides, since we know {{{sin(alpha)}}} is positive)

Using the same sort of calculation, we get that {{{sin(beta) = 5/13}}}.  So now we substitute into the sum formula:

{{{sin(alpha + beta) = sin(alpha)*cos(beta) + cos(alpha)*sin(beta)}}}
{{{sin(alpha + beta) = (3/5)*(12/13) + (4/5)*(5/13)}}} (substituting)
{{{sin(alpha + beta) = 36/65 + 20/65}}} (simplifying)
{{{sin(alpha + beta) = 56/65}}} (simplifying)