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  4. If A, B, C are in arithmetic progression and B=(π /4), then tan Atan ⁡ Btan ⁡ C=

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Q. If $A, \, B, \, C$ are in arithmetic progression and $B=\frac{\pi }{4},$ then $tan Atan ⁡ Btan ⁡ C=$

NTA AbhyasNTA Abhyas 2020

Solution:

$\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 $tan \left(\frac{\pi }{4} - \theta \right) tan ⁡ \frac{\pi }{4} tan ⁡ \left(\frac{\pi }{4} + \theta \right)$
$=\frac{1 - tan \theta }{1 + tan ⁡ \theta }.1.\frac{1 + tan ⁡ \theta }{1 - tan ⁡ \theta }$
$=1$