Question

Use the sum and difference identities to determine the exact value of the following expression. If the answer is undefined, write DNE.cos115 cos65° - sin115 sin65°

Answer 1

We can use the following identity to solve the exercise:

$\cos a\cos b-\sin a\sin b=\cos (a+b)$

In this case, we have:

$\begin{gathered} a=115\degree \\ b=65\degree \end{gathered}$$\begin{gathered} \cos a\cos b-\sin a\sin b=\cos (a+b) \\ \cos 115\degree\cos 65\degree-\sin 115\degree\sin 65\degree=\cos (115\degree+65\degree) \\ \cos 115\degree\cos 65\degree-\sin 115\degree\sin 65\degree=\cos (180\degree) \\ \cos 115\degree\cos 65\degree-\sin 115\degree\sin 65\degree=-1 \end{gathered}$

Therefore, the exact value of the given expression is -1.