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3. trignometrical ratios,functions and identities — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS3. trignometrical ratios,functions and identities1 Mark
MCQ
If $\cos \,\left( {\alpha  + \beta } \right) = \frac{3}{5},\,\sin \,\left( {\alpha  - \beta } \right) = \frac{5}{{13}}$ and $0 < \alpha ,\beta  < \frac{\pi }{4}$ then $\tan \,\left( {2\alpha } \right)$ is equal to
  • A
    $\frac{{63}}{{52}}$
  • B
    $\frac{{33}}{{52}}$
  • ✓
    $\frac{{63}}{{16}}$
  • D
    $\frac{{21}}{{16}}$
✓

Answer

Correct option: C.
$\frac{{63}}{{16}}$
c
$0\, < \,\alpha \, + \,\beta \, = \,\frac{\pi }{2}$ and $\frac{{ - \pi }}{4} < \,\alpha \, - \,\beta \, < \,\frac{\pi }{4}$

If $\cos \,(\,\alpha  + \,\beta )\, = \,\frac{3}{5}$ then $\tan \,(\,\alpha  + \,\beta )\, = \,\frac{3}{4}$ and if $\sin \,(\,\alpha  - \,\beta )\, = \,\frac{5}{{13}}$ then $\tan \,(\,\alpha  - \,\beta )\, = \,\frac{5}{{12}}$

(since $\alpha  - \,\beta $ here lies in the first quadrant)

Now $\tan \,(\,2\alpha ) = \tan \{ (\alpha \, + \,\beta ) + (\alpha \, - \,\beta )\} $

$ = \,\frac{{\tan \,(\alpha \, + \,\beta ) + \tan \,(\alpha \, - \,\beta )}}{{1 - \tan \,(\alpha \, + \,\beta ).\tan \,(\alpha \, - \,\beta )}}$ 

$ = \frac{{\frac{4}{3} + \frac{5}{{12}}}}{{1 - \frac{4}{3}.\frac{5}{{12}}}} = \frac{{63}}{{16}}$

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