1. **State the problem:** Evaluate the expression $$\sqrt{\left(4! + e^{\ln(32)}\right)^2 - 2} - \sin^2(0) + \frac{\log_2(64)}{\cos^2(0)} + 2.$$\n\n2. **Recall formulas and rules:**\n- Factorial: $4! = 4 \times 3 \times 2 \times 1 = 24$.\n- Exponent and logarithm: $e^{\ln(x)} = x$.\n- Trigonometric values: $\sin(0) = 0$, $\cos(0) = 1$.\n- Logarithm base 2: $\log_2(64)$ means the power to which 2 must be raised to get 64.\n\n3. **Calculate intermediate values:**\n- Calculate $4!$: $$4! = 24.$$\n- Calculate $e^{\ln(32)}$: $$e^{\ln(32)} = 32.$$\n- Sum inside the square: $$4! + e^{\ln(32)} = 24 + 32 = 56.$$\n- Square the sum: $$56^2 = 3136.$$\n- Subtract 2: $$3136 - 2 = 3134.$$\n- Take the square root: $$\sqrt{3134}.$$\n- Calculate $\sin^2(0)$: $$\sin(0) = 0 \Rightarrow \sin^2(0) = 0.$$\n- Calculate $\log_2(64)$: Since $2^6 = 64$, $$\log_2(64) = 6.$$\n- Calculate $\cos^2(0)$: $$\cos(0) = 1 \Rightarrow \cos^2(0) = 1.$$\n- Divide logarithm by cosine squared: $$\frac{6}{1} = 6.$$\n\n4. **Combine all parts:**\n$$\sqrt{3134} - 0 + 6 + 2 = \sqrt{3134} + 8.$$\n\n5. **Final answer:**\n$$\boxed{\sqrt{3134} + 8}.$$\n\nThis is the simplified exact form. Numerically, $\sqrt{3134} \approx 55.99$, so the value is approximately $63.99$.