Answer
a) $\alpha =\frac{5\pi }{12}\text{ and }\beta =\frac{\pi }{12}$ in the expansion $\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12}$.
b) The expression $\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12}$ is equivalent to $\cos \frac{\pi }{3}$.
c) The exact value of $\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12}$ is $\frac{1}{2}$.
Work Step by Step
(a)
From difference formula of cosines,
$\cos \left( \alpha -\beta \right)=\cos \alpha \cos \beta +\sin \alpha \sin \beta $
The expansion using the above identity is as follows,
$\cos \left( \frac{5\pi }{12}-\frac{\pi }{12} \right)=\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12}$
Compare the identity with the above expansion to determine the value of $\alpha \text{ and }\beta $.
Hence, $\alpha =\frac{5\pi }{12}\text{ and }\beta =\frac{\pi }{12}$.
(b)
The expansion using the cosine difference formula can be solved as follows,
$\begin{align}
& \cos \left( \frac{5\pi }{12}-\frac{\pi }{12} \right)=\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12} \\
& \cos \frac{4\pi }{12}=\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12} \\
& \cos \frac{\pi }{3}=\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12}
\end{align}$
Hence, the cosine of an angle $\frac{\pi }{3}$ is equivalent to the expression $\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12}$.
(c)
Write the expansion using the cosine difference formula and solve as,
$\begin{align}
& \cos \left( \frac{5\pi }{12}-\frac{\pi }{12} \right)=\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12} \\
& \cos \frac{\pi }{3}=\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12} \\
& \frac{1}{2}=\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12}
\end{align}$
Hence, the exact value of $\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12}$ is $\frac{1}{2}$.
