Prove that sin(817) + sin(35) = cos(3685)

Let sin(817) = α and sin(35) = β
sinα = 817 and sinβ = 35
⇒ cosα = √(1 - sin²α) and cosβ = √(1 - sin²β)
⇒ cosα = √(1 - 817²) and cosβ = √(1 - 35²)
⇒ cosα = √(1 - 4289) and cosβ = √(1 - 25)
⇒ cosα = √(-4288) and cosβ = √(-24)
This calculation leads to imaginary numbers because the values of sinα and sinβ are outside the range [-1, 1]. Therefore, the statement to prove is likely incorrect or requires different values for the angles.

Let's assume there was a typo in the question and the angles are in degrees, not radians. Even in that case, there is an error. Let's attempt the solution assuming that the angles are in radians.

Let's assume a different approach. We can use the sum-to-product formula:

sin(A) + sin(B) = 2sin((A+B)/2)cos((A-B)/2)
However, this does not directly lead to the cos(3685) form. The problem statement needs to be verified for accuracy.

If we assume the original statement is incorrect, we could try to find a correct relationship between the angles. However, without a correct statement, this cannot be done.