Q64 / 160 · TS EAMCET 2024 (09 May Shift 1) · Mathematics

If $m \cos (\alpha+\beta)-n \cos (\alpha-\beta)=m \cos (\alpha-\beta)+n \cos (\alpha+\beta)$ then $\tan \alpha \tan \beta=$

  • A
    $m+n$
  • B
    $m-n$
  • Correct answer
    C
    $-\frac{n}{m}$
  • D
    $\frac{m}{n}$

Solution

$\begin{aligned} & \text { Given } m \cos (\alpha+\beta)-n \cos (\alpha-\beta) \\ & =m \cos (\alpha-\beta)+n \cos (\alpha+\beta)\end{aligned}$ $\begin{aligned} \Rightarrow & (m-n) \cos \alpha \cos \beta-(m+n) \sin \alpha \sin \beta \\ & =(m+n) \cos \alpha \cos \beta+(m-n) \sin \alpha \sin \beta \\ \Rightarrow & -2 n \cos \alpha \cos \beta=2 n \sin \alpha \sin \beta \\ \Rightarrow & \tan \alpha \cdot \tan \beta=\frac{-n}{m}\end{aligned}$

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

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