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2 problems match “angle addition formula”. Clear

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1. Maximum of tan beta under a constraint ●●●○○ 50 ptsGiven $\alpha, \beta \in\left(0, \frac{\pi}{2}\right)$ with $\alpha+\beta \neq \frac{\pi}{2}$ and $\sin \beta=2 \cos (\alpha+\beta) \sin \alpha$, find the maximum value of $\tan \beta$. Round to two decimal places.
2. Sine of a shifted double angle ●●●○○ 50 ptsGiven $f(x)=2\cos \left(x-\frac{\pi}{3}\right)\cos x+1$, and $f\left(\alpha +\frac{\pi}{12}\right)=\frac{7}{6}$ with $\alpha \in \left(0,\frac{\pi}{2}\right)$, find $\sin \left(\frac{7\pi}{6}-2\alpha \right)$. Round to two decimal places.