Establish the identity. \[ \frac{4 \cot u-\tan u}{\cot u+\tan u}+1=5 \cos ^{2} u \]

To establish the identity \[ \frac{4 \cot u - \tan u}{\cot u + \tan u} + 1 = 5 \cos^2 u, \] we start by rewriting \(\cot u\) and \(\tan u\) in terms of sine and cosine: \[ \cot u = \frac{\cos u}{\sin u}, \quad \tan u = \frac{\sin u}{\cos u}. \] Substituting these into the left-hand side of the identity gives: \[ \frac{4 \cot u - \tan u}{\cot u + \tan u} = \frac{4 \frac{\cos u}{\sin u} - \frac{\sin u}{\cos u}}{\frac{\cos u}{\sin u} + \frac{\sin u}{\cos u}}. \] To simplify this expression, we find a common denominator for both the numerator and the denominator. **Numerator:** \[ 4 \frac{\cos u}{\sin u} - \frac{\sin u}{\cos u} = \frac{4 \cos^2 u - \sin^2 u}{\sin u \cos u}. \] **Denominator:** \[ \frac{\cos u}{\sin u} + \frac{\sin u}{\cos u} = \frac{\cos^2 u + \sin^2 u}{\sin u \cos u} = \frac{1}{\sin u \cos u} \quad \text{(since } \cos^2 u + \sin^2 u = 1\text{)}. \] Now substituting these back into the fraction: \[ \frac{4 \cot u - \tan u}{\cot u + \tan u} = \frac{\frac{4 \cos^2 u - \sin^2 u}{\sin u \cos u}}{\frac{1}{\sin u \cos u}} = 4 \cos^2 u - \sin^2 u. \] Now we can rewrite the left-hand side of the original identity: \[ 4 \cos^2 u - \sin^2 u + 1. \] Using the Pythagorean identity \(\sin^2 u = 1 - \cos^2 u\), we can substitute for \(\sin^2 u\): \[ 4 \cos^2 u - (1 - \cos^2 u) + 1 = 4 \cos^2 u - 1 + \cos^2 u + 1 = 5 \cos^2 u. \] Thus, we have shown that: \[ \frac{4 \cot u - \tan u}{\cot u + \tan u} + 1 = 5 \cos^2 u. \] Therefore, the identity is established: \[ \boxed{5 \cos^2 u}. \]