Question

If $\tan \left(A-B\right)=1\sqrt{3}$ and $tan\left(A+B\right)=\sqrt{3}$ , find A and B.

Answer 1

Consider the given trigonometric value

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

$\tan (A-B)=\tan \frac{\pi }{6},\tan (A+B)=\tan \frac{\pi }{3}$

$A-B=\frac{\pi }{6}....\left(1\right),A+B=\frac{\pi }{3}.......\left(2\right)$

From equation 1^st and 2^nd we get,

$2A=\frac{\pi }{3}+\frac{\pi }{6}=\frac{3\pi }{6}=\frac{\pi }{6}$

$A=\frac{\pi }{12}$

Put the value of A in equation 1^st we get.

$\frac{\pi }{12}-B=\frac{\pi }{6}$

$B=\frac{\pi }{12}+\frac{\pi }{6}=\frac{3\pi }{12}=\frac{\pi }{4}$

Hence,$A=\frac{\pi }{12},B=\frac{\pi }{4}$ are the required answer ..