First slide

Compound angles

Question

If $\cos (α + β) = 4 / 5$ and $\sin (α - β) = 5 / 13$ where $0 \leq α, β \leq π / 4$ , then $\tan 2 α =$

Moderate

A

$\frac{19}{12}$
B

$\frac{20}{17}$
C

$\frac{25}{16}$
D

$\frac{56}{33}$

Solution

$0 \leq α, β \leq π / 4$
$\Rightarrow 0 \leq α + β \leq π / 2 \Rightarrow - π / 4 \leq α - β \leq π / 4$
Now $\cos (α + β) = 4 / 5 \Rightarrow \tan (α + β) = 3 / 4$
and $\sin (α - β) = 5 / 13 \Rightarrow \tan (α - β) = 5 / 12$
we have $\tan 2 α = \tan [(α + β) + (α - β)]$
$\begin{array}{l} = \frac{\tan (α + β) + \tan (α - β)}{1 - \tan (α + β) \tan (α - β)} \\ = \frac{(3 / 4) + (5 / 12)}{1 - (3 / 4) (5 / 12)} = \frac{14 / 12}{33 / 48} = \frac{56}{33} . \end{array}$

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