Hi folks,

I am trying to find the general solution to the following trig equation:

$tan 7\theta - tan(3\theta + \frac{\pi}{4}) = 0$

$\dfrac{sin 7\theta}{cos 7\theta} - \dfrac{sin(3\theta + \frac{\pi}{4})}{cos (3\theta + \frac{\pi}{4})} = 0$

$\dfrac{ sin 7\theta. cos (3\theta + \frac{\pi}{4}) - cos 7\theta. sin(3\theta + \frac{\pi}{4}) }{cos 7\theta. cos (3\theta + \frac{\pi}{4})} = 0$

Now, the numerator simplifies quite nicely to $sin (4\theta - \frac{\pi}{4}) = 0$ and it's quite easy to get the general solution. My question to the forum is about the denominator.

In equations with $\dfrac{numerator}{denominator} = 0$ we usually throw the denominator away (because it is zero) and then work only with the numerator and we find solutions for it.

If the numerator were, for example, $sin 4\theta. cos \theta = 0$ we would say $sin 4\theta = 0$ OR $cos \theta = 0$ and provide two sets of solutions.

So why don't we consider $cos 7\theta. cos(3\theta + \frac{\pi}{4}) = 0$?

thanks