write the product as a sum: 10 cos(37q) sin(18q) =

Explanation:

Step1: Use product - to - sum formula

We use the formula $2\cos A\sin B=\sin(A + B)-\sin(A - B)$. Here $A = 37q$ and $B=18q$, and the given expression is $10\cos(37q)\sin(18q)$. First, rewrite it as $5\times(2\cos(37q)\sin(18q))$.

Step2: Substitute values into formula

Substitute $A = 37q$ and $B = 18q$ into $2\cos A\sin B=\sin(A + B)-\sin(A - B)$. We get $5\times(\sin(37q+18q)-\sin(37q - 18q))$.

Step3: Simplify the expressions inside sine functions

Calculate $37q+18q = 55q$ and $37q - 18q=19q$. So the expression becomes $5\sin(55q)-5\sin(19q)$.

Answer:

$5\sin(55q)-5\sin(19q)$