If ( + ) = 4 5 and ( - ) = 5 13 ,where 0 , 4 . Then 2 = - Vidyadip

MCQ

If $\cos \left( {\alpha + \beta } \right) = \frac{4}{5}$ and $\sin \left( {\alpha - \beta } \right) = \frac{5}{{13}}$,where $0 \le \alpha ,\beta \le \frac{\pi }{4}$ . Then $\tan 2\alpha =$

A
$\frac{{16}}{{63}}$
✓
$\frac{{56}}{{33}}$
C
$\frac{{28}}{{33}}$
D
None of these

✓

Answer

Correct option: B.

$\frac{{56}}{{33}}$

b
$\cos \,(\alpha \, + \beta ) = \frac{4}{5} \Rightarrow \tan (\alpha \, + \beta ) = \frac{3}{4}$

$\sin \,(\alpha \, - \beta )\, = \frac{5}{{13}} \Rightarrow \tan \,(\alpha \, - \beta ) = \frac{5}{{12}}$

$\tan 2\alpha \, = \tan [(\alpha \, + \beta ) + (\alpha \, - \beta )]$ $\, = \,\frac{{\frac{3}{4} + \frac{5}{{12}}}}{{1 - \frac{3}{4}.\frac{5}{{12}}}} = \frac{{56}}{{33}}$

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