Question

tan theta + tan 4 theta + tan 7 theta = tan theta tan 4 theta tan 7 theta , then theta = answer with explanation ​

Answer 1

Step-by-step explanation:

Hint

**8x = x + 7x = 4x + 4x

$\tan(x) + \tan(4x) + \tan(7x) - \tan(x) \tan(4x) \tan(7x) = 0 \\ \tan(x) + \tan(7x) = \tan(4x) (\tan(x) \tan(7x) - 1)$

Now use

$\tan(8x) = \tan(8x) \\ \tan(2 \times 4x) = \tan(x + 7x ) \\ \\ \frac{2 \tan(4x) }{1 - \tan {}^{2} (4x) } = \frac{\tan(x) + \tan(7x)}{1 - \tan(x) \tan(7x)} \\ use \: \: above \: \: result \\ \\ \frac{2 \tan(4x) }{1 - \tan {}^{2} (4x) } = \frac{\tan(4x) ( \tan(x) \tan(7x) - 1)}{1 - \tan(x) \tan(7x)} \\ \\ 2 \tan(4x) = - \tan(4x) (1 - { \tan }^{2}4 x) \\ \tan(4x) ( { \tan}^{2} 4x - 3) = 0 \\ \tan(4x) = 0 \: \: and \: \: \tan(4x) = \pm \sqrt{3} \\ 4x = n\pi \: \: and \: \: n\pi \: \pm \: \frac{\pi}{3} \\ x = \frac{n\pi}{4} \: \: and \: \: \frac{1}{12} (3n \pi \: \pm \: \pi)$