Q157 / 270 · NDA 2011 (Phase 1) · Mathematics

If $(1+\tan \theta)(1+\tan \varphi)=2$, then what is $(\theta+\varphi)$ equal to?

  • A
    $30^{\circ}$
  • Correct answer
    B
    $45^{\circ}$
  • C
    $60^{\circ}$
  • D
    $90^{\circ}$

Solution

$\begin{aligned} & \text { Given }(1+\tan \theta)(1+\tan \phi)=2 \\ & \Rightarrow \quad 1+\tan \theta+\tan \phi+\tan \theta \cdot \tan \phi=2 \\ & \Rightarrow \quad \tan \theta+\tan \phi=1-\tan \theta \cdot \tan \phi \\ & \Rightarrow \quad \frac{\tan \theta+\tan \phi}{1-\tan \theta \cdot \tan \phi}=1 \\ & \Rightarrow \quad \tan (\theta+\phi)=\tan \frac{\pi}{4} \\ & \Rightarrow \quad \theta+\phi=\frac{\pi}{4}=45^{\circ} .\end{aligned}$

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

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