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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 $\tan \alpha = {(1 + {2^{ - x}})^{ - 1}},$ $\tan \beta = {(1 + {2^{x + 1}})^{ - 1}}$, then $\alpha + \beta $ equals
  • A
    $\pi /6$
  • ✓
    $\pi /4$
  • C
    $\pi /3$
  • D
    $\pi /2$
✓

Answer

Correct option: B.
$\pi /4$
b
(b) $\tan \,(\alpha + \beta ) = \frac{{\tan \alpha + \tan \beta }}{{1 - \tan \alpha \tan \beta }}$

==> $\tan (\alpha + \beta ) = \frac{{\frac{1}{{1 + \frac{1}{{{2^x}}}}} + \frac{1}{{1 + {2^{x + 1}}}}}}{{1 - \frac{1}{{1 + 1/{2^x}}}\frac{1}{{1 + {2^{x + 1}}}}}}$

==> $\tan (\alpha + \beta ) = \frac{{{2^x} + {{2.2}^{x + x}} + {2^x} + 1}}{{1 + {2^x} + {{2.2}^x} + {{2.2}^{x + x}} - {2^x}}}$

==> $\tan (\alpha + \beta ) = 1$

$ \Rightarrow \alpha + \beta = \frac{\pi }{4}$.

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