<s>[INST] Solve a given trigonometric equation Solve the following equation:
$$\sin x + \sin \left( x + \frac{7\pi}{24} \right) = \frac{\sqrt{2 - \sqrt 2}}{2} + \frac{\sqrt 6 + \sqrt 2}{4}$$
So far, I found out that $\frac{\sqrt{2 - \sqrt 2}}{2} = \sin \frac{\pi}{8}$ and $\frac{\sqrt 6 + \sqrt 2}{4} = \sin \frac{7\pi}{12}$.
Thank you!
 [/INST] Hint
Expand the second $\sin$, to write your 
equation as
$$A\cos(x)+B\sin(x)=1$$
or
$$\cos(x+\alpha)=\frac{1}{\sqrt{A^2+B^2}}$$

second approach

the equation is
$$
\sin(x)+\sin(x+\frac{7\pi}{24})=$$  
$$\sin( \frac{\pi}{8})+\sin(\frac{7\pi}{12}  )=$$
$$\sin(\frac{\pi}{8})+\sin\left(\pi-(\frac{\pi}{8}+\frac{7\pi}{24})\right)=$$
$$\sin(\frac{\pi}{8})+\sin(\frac{\pi}{8}+\frac{7\pi}{24})$$
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