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If $\tan A=1 / 2$ and $\tan B=1 / 3$, then what is the value of $4 A+4 B ?$
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Verified Answer
The correct answer is:
$\pi$
Let tan $A=\frac{1}{2}, \tan B=\frac{1}{3}$
We know,
$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2} \times \frac{1}{3}}=\frac{5}{6} \times \frac{6}{5}$
$\tan (A+B)=1$
$\Rightarrow A+B=\tan ^{-1}(1)=\frac{\pi}{4}$
Multiply by 4 on both side,
$4(A+B)=\frac{\pi}{4} \times 4 \Rightarrow 4 A+4 B=\pi$
We know,
$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2} \times \frac{1}{3}}=\frac{5}{6} \times \frac{6}{5}$
$\tan (A+B)=1$
$\Rightarrow A+B=\tan ^{-1}(1)=\frac{\pi}{4}$
Multiply by 4 on both side,
$4(A+B)=\frac{\pi}{4} \times 4 \Rightarrow 4 A+4 B=\pi$
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