If (+)=35, (-)=513, and 0

Question

If $\cos(\alpha+\beta)=\dfrac35$, $\sin(\alpha-\beta)=\dfrac5{13}$, and $0<\alpha,\beta<\dfrac\pi4$, then $	an(2\alpha)$ is equal to:

Accepted Answer

Since $\alpha+\beta,\alpha-\beta$ are both acute: $\sin(\alpha+\beta)=\dfrac45 \implies \tan(\alpha+\beta)=\dfrac43$; $\cos(\alpha-\beta)=\dfrac{12}{13}\implies\tan(\alpha-\beta)=\dfrac5{12}$.

$$\tan2\alpha = \frac{\tan(\alpha+\beta)+\tan(\alpha-\beta)}{1-\tan(\alpha+\beta)\tan(\alpha-\beta)} = \frac{\tfrac43+\tfrac5{12}}{1-\tfrac43\cdot\tfrac5{12}} = \frac{63/16}1 \cdot\frac1{1} = \frac{63}{16}$$

Solutions · 1