1. Identidades a usar:

• $\sec \theta = \frac{1}{\cos \theta}$
• $\cos 2x = 2\cos^2 x - 1$
• $\cos 3x = \cos x(4\cos^2 x - 3)$
• $\tan 2x = \frac{2\tan x}{1 - \tan^2 x}$

2. Simplificación por términos:

• Término 1: $\frac{\sec^2 x}{\sec 2x} = \frac{\cos 2x}{\cos^2 x} = \frac{2\cos^2 x - 1}{\cos^2 x} = 2 - \sec^2 x = 1 - \tan^2 x$
• Término 2: $\frac{\sec^3 x}{\sec 3x} = \frac{\cos 3x}{\cos^3 x} = \frac{4\cos^3 x - 3\cos x}{\cos^3 x} = 4 - 3\sec^2 x = 4 - 3(1 + \tan^2 x) = 1 - 3\tan^2 x$
• Término 3: $- \frac{8\tan x}{\tan 2x} = - 8\tan x \left( \frac{1 - \tan^2 x}{2\tan x} \right) = -4(1 - \tan^2 x) = 4\tan^2 x - 4$

3. Suma total:
$$A = (1 - \tan^2 x) + (1 - 3\tan^2 x) + (4\tan^2 x - 4)$$
$$A = (1 + 1 - 4) + (-1 - 3 + 4)\tan^2 x$$
$$A = -2 + 0$$
Resultado: $A = -2$