since the cosine is the adjacent over the hypotenuse, 
and we are given that the cosine of x is 8 over 17, we 
will draw a right triangle with an adjacent side of 8
and a hypotenuse of 17.  Then the angle in that 
triangle that has 8 for its adjacent side will be 
angle x.

Since the tangent is the opposite over the adjacent, 
and we are given that the tangent of y is 5 over 12, 
we will draw a right triangle with an opposite side 
of 5 and an adjacent side of 12.  Then the angle in 
that triangle that has 5 for its opposite side and 
12 for its adjacent side will be angle y.

 

Now we will find the missing side in each of those
triangles using the Pythagorean theorem:

 c² = a² + b²         c² = a² + b²
17² = 8² + b²         c² = 12² + 5²
289 = 64 + b²         c² = 144 + 25
225 = b²              c² = 169
 15 = b                c = 13



Now we use the identity

sin(x+y) = sin(x)cos(y) + cos(x)sin(y)

and use the sides of those triangles to substitute
the trig ratios:

sin(x+y) =  =  =  

Edwin