Mad Trigonometry

Formula used :

$\large \cos(3x) = 4 \cos ^ 3 (x) - 3 \cos(x)$

$\large \cos^2(x) = \frac{1+\cos(2x)}{2}$

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$\large \dfrac{\cos(756x)}{\cos(252x)}-\dfrac{\cos(1512x)}{\cos(504x)} =2$

$\large \dfrac{4 \cos ^ 3 (252x) - 3 \cos(252x)}{\cos(252x)}-\dfrac{4 \cos ^ 3 (504x) - 3 \cos(504x)}{\cos(504x)} =2$

$\large 4 \cos^2(252x) -4 \cos^2(504x) -3+3 = 2$

$\large -4 \cos^2(504x)+2\cos(504x)+2=2$

$\large -4 \cos^2(504x)+2\cos(504x) = 0$

$\large \cos(504x) = 0$ or $\large \cos(504x) = \frac{1}{2}$

Observe patterns :

The sum of roots of $\large \cos(504x) = 0$ for $0 < x < 2\pi$ = $2(504)$ = $1008$

The sum of roots of $\large \cos(504x) = \frac{1}{2}$ for $0 < x < 2\pi$ = $2(504)$ = $1008$

$\large \Rightarrow 1008+1008 = 2016$