Question

How do you calculate the exact value of cos(u+v) given cos u and sin v?

Answer 1

we have to apply this identity:
\[\cos \left( {u + v} \right) = \cos \left( u \right)\cos \left( v \right) - \sin \left( u \right)\sin \left( v \right)\]

Answer 2

furthermore, we have to keep in mind that:
\[\begin{gathered}
{\left( {\sin u} \right)^2} + {\left( {\cos u} \right)^2} = 1 \hfill \\
{\left( {\sin v} \right)^2} + {\left( {\cos v} \right)^2} = 1 \hfill \\
\end{gathered} \]