The expression you provided is quite complex, with a mix of nested functions, trigonometric operations, logarithms, roots, imaginary components, and constants. To make progress with simplifying and evaluating this expression, we need to break it down into smaller, manageable components.

Here are steps to simplify and analyze such an expression systematically:

Step 1: Identify Key Subcomponents

1. Constants and approximations:

   • $\pi \approx 3.14159$
   • $e \approx 2.71828$
   • $i$ is the imaginary unit ($i^2 = -1$).

2. Outer structure:

   • Functions like $\ln$, $\cos$, $\tan$, and nested roots require simplifying inside-out.
   • There is frequent use of the operator $\angle$, which could be interpreted as composing multiple operations or using vector notation.

Step 2: Simplify Approximations

• Exponents and logarithmic terms can often be approximated step by step.
• Evaluate constants like $\ln(1) = 0$, $\tan(\pi) = 0$, $\cos(0.2229)$, and similar values numerically.

Step 3: Handle Nested Roots and Logs

• Terms like $\text{root}(\tan(\cos(0.043001)))$ and $\ln(\ln(\text{root}(e)))$ need careful isolation and computation.

Step 4: Complex Numbers

• Any terms involving $i$ will form part of the complex plane and should be treated with Euler's formula when possible: $e^{i\theta} = \cos(\theta) + i\sin(\theta).$

Would you like me to simplify and numerically approximate this entire expression or focus on a specific part?