Let (+)=45 and (-)=513 where 0,4.

Question

Let $\cos(\alpha+\beta)=\dfrac45$ and $\sin(\alpha-\beta)=\dfrac5{13}$ where $0\le\alpha,\beta\le\dfrac\pi4$. Then $	an2\alpha=$

Accepted Answer

$$\cos(\alpha+\beta)=\frac45 \implies \tan(\alpha+\beta)=\frac34, \qquad \sin(\alpha-\beta)=\frac5{13}\implies\tan(\alpha-\beta)=\frac5{12}$$

$$\tan2\alpha = \tan[(\alpha+\beta)+(\alpha-\beta)] = \frac{\tfrac34+\tfrac5{12}}{1-\tfrac34\cdot\tfrac5{12}} = \frac{14/12}{33/48}\cdot(\text{simplify}) = \frac{56}{33}$$

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