Hard Trigonometric Equation

Question

its possible to solve a equation like

$$\prod^{45}_{k = 0} \left( 1 + \tan \frac{k \pi }{180} \right) = \left[ \log_{\frac{\sqrt{6}}{3}} \big| \sin(2x)\big| \right]^{\frac{9}{2}\sin(2x) + 20}$$

where I got this equation was wondering which of the intervals contained a solution of it?

1. $[0,631|0,747]$
2. $[0,258|0,395]$
3. $[0,412|0,627]$
4. $[0,799|0,814]$
5. $[0,041|0,107]$

Answer 1

Following the link by lab you get that the product of the tan's is $2^{23}$ if you set

$$\frac{9}{2}\sin (2x)+20=23$$ and

$$\log_{\frac{\sqrt{6}}{3}}\sin(2x)=2$$ you find in both cases that $$\sin(2x)=\frac{2}{3}$$