If cos = 4 5 and sin = 5 13 , where 0 , 4 , then tan 2 is equal to

Question

If $\cos (α + β) = \frac{4}{5}$ and $\sin (α - β) = \frac{5}{13},$ where $0 \leq α, β \leq \frac{π}{4},$ then $\tan 2 α$ is equal to

Smart Achievers · Accepted Answer

(2)

$\cos (α + β) = \frac{4}{5}$

$\Rightarrow$ $α + β \in 1st quadrant$

$\sin (α - β) = \frac{5}{13}$

$\Rightarrow$ $(α - β) \in 1st quadrant$

Now, $2 α = (α + β) + (α - β)$

$∴$ $\tan 2 α = \frac{\tan (α + β) + \tan (α - β)}{1 - \tan (α + β) \tan (α - β)}$

$= \frac{\frac{3}{4} + \frac{5}{12}}{1 - \frac{3}{4} \cdot \frac{5}{12}} = \frac{\frac{56}{48}}{\frac{33}{48}} = \frac{56}{33}$

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