Starting with \(5(\tan^2 x - \cos^2 x) = 2\cos 2x + 9\). We have \(\tan^2 x = \frac{\sin^2 x}{\cos^2 x}\), so \(5(\frac{\sin^2 x}{\cos^2 x} - \cos^2 x) = 2\cos 2x + 9\). Multiplying through by \(\cos^2 x\): \(5\sin^2 x - 5\cos^4 x = 2\cos 2x \cos^2 x + 9\cos^2 x\). This becomes complex. Alternatively, using \(\cos 2x = 2\cos^2 x - 1 = 1 - 2\sin^2 x\) and manipulating: after algebraic simplification, we get \(\cos 4x = -\frac{7}{9}\).