Q60 / 160 · AP EAMCET 2023 (18 May Shift 1) · Mathematics

If $\tan \mathrm{B}=\frac{2 \sin \mathrm{A} \sin \mathrm{C}}{\sin (\mathrm{A}+\mathrm{C})}$, then $\tan \mathrm{A}, \tan \mathrm{B}$ and $\tan \mathrm{C}$ are in

  • A
    Arithmetic progression
  • Correct answer
    B
    Harmonic progression
  • C
    Geometric progression
  • D
    Arithmetico - geometric progression

Solution

$\tan B=\frac{2 \sin A \sin C}{\sin (A+C)} \Rightarrow \frac{1}{\tan B}=\frac{\sin (A+C)}{2 \sin A \sin C}$ $\Rightarrow \frac{2}{\tan B}=\frac{1}{\tan C}+\frac{1}{\tan A}$ $\therefore \quad \tan \mathrm{A}, \tan \mathrm{B} \& \tan \mathrm{C}$ are in harmonic progression.

Hence, the correct answer is option: 'B'.

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