Sink Cost Tank Cots

Geometry Level 5

\[\dfrac{\sin^{2y}x}{\cos^{\frac{y^2}{2}}x} + \dfrac{\cos^{2y}x}{\sin^{\frac{y^2}{2}}x} = \dfrac{2\tan x}{1+\tan^2x}\]

There are \( n\) pairs of real numbers \(x,y\), such that \(\ x \in \Big( 0, \dfrac{\pi}{2} \Big) \) , that satisfy the above equation.
Let \(\alpha=\dfrac{4}{3} (x_1y_1 + x_2y_2 + \cdots + x_ny_n)\)

Then the value of\((\sin \alpha + \sin 4\alpha + \sin 7\alpha + \cdots + \sin 298\alpha)\) can be expressed as \(a \sqrt b \), where \(a\) and \(b\) are positive integers with \(b\) is square free. Find \(a+b\).


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