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Trigonometric Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Functions1 Mark
MCQ
Choose the correct answer. If $\tan\text{A}=\frac{1}{2},\tan\text{B}=\frac{1}{3},$ then $\tan(2\text{A + B})$ is equal to:
  • A
    $1$
  • B
    $2$
  • ✓
    $3$
  • D
    $4$
✓

Answer

Correct option: C.
$3$
Given that, $\tan\text{A}=\frac{1}{2}$ and $\tan\text{B}=\frac{1}{3}$
$\tan2\text{A}=\frac{2\tan\text{A}}{1-\tan^2\text{A}}=\frac{2\times\frac{1}{2}}{1-\Big(\frac{1}{2}\Big)^2}$
$=\frac{1}{1-\frac{1}{4}}=\frac{1}{\frac{3}{4}}=\frac{4}{3}$
so, $\tan2\text{A}=\frac{4}{3}$ and $\tan\text{B}=\frac{1}{3}$
$\tan(\text{2A+B})=\frac{\tan2\text{A}+\tan\text{B}}{1-\tan\text{A}.\tan\text{B}}=\frac{\frac{4}{3}+\frac{1}{3}}{1-\frac{4}{3}\times\frac{1}{3}}$
$=\frac{\frac{5}{3}}{\frac{9-4}{9}}=\frac{5}{3}\times\frac{9}{5}=3$
Hence, the correct option is $(c).$

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