Trigon and Metron

I used sum of product identities to solve this question. ( Proofs of these formula here ). The formulas I used is as follows:

1. $sinx - sin y = 2cos(\frac{x+y}{2}) \cdot sin(\frac{x-y}{2})$
2. $cosx + cos y = 2cos(\frac{x+y}{2}) \cdot cos(\frac{x-y}{2})$
3. $cosx - cos y = -2sin(\frac{x+y}{2}) \cdot sin(\frac{x-y}{2})$

Notice that the question simplifies into:

$\frac{ \frac{sin 7x - sinx -(sin3x - sinx)}{2}}{\frac{-(cos 5x - cos3x) +cos7x +cos5x}{2}} = \frac{sin 7x - sin3x}{cos7x+cos3x}$

Using the above formula 1 and 2 , the expression becomes: $\frac{2cos5x \cdot sin2x}{2cos5x \cdot cos2x} = \frac{sin 2x}{cos2x} = \boxed{tan2x}$

The trololol solution:

Plugging in x=Pi, we see that only 1 and 2 work.

Plugging in x=Pi/2, we get that only 2 works. QED

there is a much easier way, though elementary SUBSTITUTE X=(pie)/2 and check the options