Question

Solve the trigonometric equation : tan theta + tan 4 theta + tan 7 theta = tan theta * tan 4 theta * tan 7 theta

Answer 1

Know we know that
$\tan( \alpha + \beta + \gamma ) = \frac{ \tan( \alpha ) + \tan( \beta ) + \tan( \gamma ) - \tan( \alpha ) \tan( \beta ) \tan( \gamma ) }{1 - \tan( \alpha ) \tan( \beta ) - \tan( \beta ) \tan( \gamma ) - \tan( \gamma ) \tan( \alpha ) }$
Now according to question,
$\tan( \alpha ) + \tan( \beta ) + \tan( \gamma ) = \tan( \alpha ) \tan( \beta ) \tan( \gamma )$
Thus,
$\tan( \alpha + \beta + \gamma ) = 0 \\ \alpha + \beta + \gamma = n\pi \\ x + 4x + 7x = n\pi \\ x = \frac{n\pi}{12}$
:)

Answer 2

Answer:

Know we know that

\tan( \alpha + \beta + \gamma ) = \frac{ \tan( \alpha ) + \tan( \beta ) + \tan( \gamma ) - \tan( \alpha ) \tan( \beta ) \tan( \gamma ) }{1 - \tan( \alpha ) \tan( \beta ) - \tan( \beta ) \tan( \gamma ) - \tan( \gamma ) \tan( \alpha ) }

Now according to question,

\tan( \alpha ) + \tan( \beta ) + \tan( \gamma ) = \tan( \alpha ) \tan( \beta ) \tan( \gamma )

Thus,

\tan( \alpha + \beta + \gamma ) = 0 \\ \alpha + \beta + \gamma = n\pi \\ x + 4x + 7x = n\pi \\ x = \frac{n\pi}{12}