# Thread: To Find the Value of tan

1. ## To Find the Value of tan

If $\cos (\alpha + \beta) = \frac{4}{5}, \sin(\alpha - \beta) = \frac{5}{13}$ and $\alpha, \beta \in (0, \frac{\pi}{4})$, then find the value of $\tan (2 \alpha)$.

2. Originally Posted by kjchauhan

If $\cos (\alpha + \beta) = \frac{4}{5}, \sin(\alpha - \beta) = \frac{5}{13}$ and $\alpha, \beta \in (0, \frac{\pi}{4})$, then find the value of $\tan (2 \alpha)$.
if $\cos(\alpha + \beta) = \frac{4}{5}$ , then $\sin(\alpha + \beta) = \frac{3}{5}$

if $\sin(\alpha - \beta) = \frac{5}{13}$ , then $\cos(\alpha - \beta) = \frac{12}{13}$

$\sin(2\alpha) = \sin[(\alpha + \beta) + (\alpha - \beta)] = \sin(\alpha + \beta)\cos(\alpha - \beta) + \cos(\alpha + \beta)\sin(\alpha - \beta)
$

$\cos(2\alpha) = \cos[(\alpha + \beta) + (\alpha - \beta)] = \cos(\alpha + \beta)\cos(\alpha - \beta) - \sin(\alpha + \beta)\sin(\alpha - \beta)$

finally, note that ...

$\displaystyle \tan(2\alpha) = \frac{\sin(2\alpha)}{\cos(2\alpha)}$

3. Thank you very much..