Question

Would just like to make sure that my answer is correct.

Answer 1

Answer:

$\text{ -2sin(}\frac{11\pi}{24})\cos (\frac{\pi}{24})$

Explanation:

Here, we want to simplify the given expression

The basic rule we will be using here is:

$\sin (A\text{ + B})\text{ = SinACosB + CosASinB}$

Thus, we have it that:

$\begin{gathered} \text{ sin(}\frac{\pi}{6}+\frac{\pi}{4})\text{ + sin(}\frac{\pi}{8}+\frac{3\pi}{8}) \\ \\ \sin (\frac{5\pi}{12})\text{ + sin(}\frac{\pi}{2}) \end{gathered}$

We use the sine addition formula as follows:

$\sin \text{ A + sin B = 2sin(}\frac{A+B}{2})\cos (\frac{A-B}{2})$

Now, we substitute the last expression into the given addition formula above:

$\begin{gathered} \text{ sin(}\frac{5\pi}{12})\text{ + sin(}\frac{\pi}{2})\text{ =2sin(}\frac{\frac{5\pi}{12}+\frac{\pi}{2}}{2})\cos (\frac{\frac{5\pi}{12}-\frac{\pi}{2}}{2}) \\ \\ =\text{ 2sin(}\frac{11\pi}{24})\cos (\frac{-\pi}{24})\text{ = -2sin(}\frac{11\pi}{24})\cos (\frac{\pi}{24}) \end{gathered}$