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Trigonometric Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Functions1 Mark
MCQ
If $\tan\alpha=\frac{\text{m}}{\text{m}+1},\tan\beta=\frac{1}{2\text{m}+1},$ then $\alpha+\beta$ is equal to:
  • A
    $\frac{\pi}{2}$
  • B
    $\frac{\pi}{3}$
  • C
    $\frac{\pi}{6}$
  • ✓
    $\frac{\pi}{4}$
✓

Answer

Correct option: D.
$\frac{\pi}{4}$
Given that, $\tan\alpha=\frac{\text{m}}{\text{m}+1}$ and $\tan\beta=\frac{1}{2\text{m}+1}$
Now, $\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\cdot\tan\beta}$
$=\frac{\frac{\text{m}}{\text{m}+1}+\frac{1}{2\text{m}+1}}{1-\Big(\frac{\text{m}}{\text{m}+1}\Big)\Big(\frac{1}{2\text{m}+1}\Big)}$
$=\frac{\text{m}(2\text{m}+1)+\text{m}+1}{(\text{m}+1)(2\text{m}+1)-\text{m}}$
$=\frac{2\text{m}^2+2\text{m}+1}{2\text{m}^2+3\text{m}+1-\text{m}}=1$
$\therefore\alpha+\beta=\frac{\pi}{4}$

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