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If $(1+\tan \theta)(1+\operatorname{tan\phi})=2$, then what is $(\theta+\phi)$ equal to?
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Verified Answer
The correct answer is:
45^{\circ}
Given, $(1+\tan \theta)(1+\tan \phi)=2$
$\Rightarrow 1+\tan \theta+\tan \phi+\tan \theta \tan \phi=2$
$\Rightarrow \tan \theta+\tan \phi=1-\tan \theta \tan \phi$
$\Rightarrow \frac{\tan \theta+\tan \phi}{1-\tan \theta \tan \phi}=1$
$\Rightarrow \tan (\theta+\phi)=\tan 45^{\circ}$
$\Rightarrow \quad \theta+\phi=45^{\circ}$
$\Rightarrow 1+\tan \theta+\tan \phi+\tan \theta \tan \phi=2$
$\Rightarrow \tan \theta+\tan \phi=1-\tan \theta \tan \phi$
$\Rightarrow \frac{\tan \theta+\tan \phi}{1-\tan \theta \tan \phi}=1$
$\Rightarrow \tan (\theta+\phi)=\tan 45^{\circ}$
$\Rightarrow \quad \theta+\phi=45^{\circ}$
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