Question

If $A,B,C$ are in arithmetic progression and $B=\frac{\pi }{4},$ then $tanAtanBtanC=$

Answer 1

Answer: 1

$\because$ Angles $A,B,C$ are in arithmetic progression and $\angle B=\frac{\pi }{4}$
then $A=\frac{\pi }{4}-\theta )tan\frac{\pi }{4}tan(\frac{\pi }{4}+\theta )$
Hence $tan\left(\frac{\pi }{4}-\theta \right)tan\frac{\pi }{4}tan\left(\frac{\pi }{4}+\theta \right)$
$=\frac{1-tan\theta }{1+tan\theta }.1.\frac{1+tan\theta }{1-tan\theta }$
$=1$