Question

If tan(A + B) = $\surd 3$ and tan(A - B) = $\frac{1}{\sqrt{3}}.$ Find A and B where A & B are acute angles.

Answer 1

$\tan (A+B)=\sqrt{3}$

$\Rightarrow \tan (A+B)=\tan {60}^{\circ }$

$\Rightarrow A+B={60}^{\circ }$ .......$\left(1\right)$

$\tan (A-B)=\frac{1}{\sqrt{3}}$

$\Rightarrow \tan (A-B)=\tan {30}^{\circ }$

$\Rightarrow A-B={30}^{\circ }$ .......$\left(2\right)$

$\left(1\right)+\left(2\right)$

$A+B+A-B={60}^{\circ }+{30}^{\circ }={90}^{\circ }$

$\Rightarrow 2A={90}^{\circ }$

$\Rightarrow A=\frac{{90}^{\circ }}{2}={45}^{\circ }$

Put $A={45}^{\circ }$ in $\left(1\right)$ we get

$B={60}^{\circ }-A={60}^{\circ }-{45}^{\circ }={15}^{\circ }$

$\therefore A={45}^{\circ },B={15}^{\circ }$