Newton Raphson to solve set of equations

I have a set of equations:

1. cot(t_{ij}/2) = p* ([cot(k_i/2)]-[cot(k_j/2)]) / ([1+p]-[1-p]*cot(k_i/2)*cot(k_j/2))
2. N*k_i = 2*pi*gamma_i + sum_over_j (t_{ij})

inputs:

p = real positive number

N = some real integer.

Indices:

• i and j can be {1,2,3...N/2}
• gamma_i is given as: {2*i-1}
• t_{ij} is actually theta_ij, i and j are two indices which are depending on N.

I want to solve this set of equations using Newton Raphson method by taking initial guess for k_i as {pi/2 + pi/N* (2*i-1)}

Example

To clarify the equations I will like to write equation explicitly for N=4 and p=1 case:

i=j={1,2}; initial guesses for k_is= {3pi/4, 5pi/4}

2 main equations will be:

cot(t_{11}/2) = 1/2 * ( cot(k_1/2) - cot(k_1/2))

cot(t_{12}/2) = 1/2 * (cot(k_1/2) - cot(k_2)/2))

cot(t_{21}/2) = 1/2 * (cot(k_2/2) - cot(k_1)/2))

cot(t_{22}/2) = 1/2 * (cot(k_2/2) - cot(k_2)/2))

N*k_1 = 2pi*gamma_1 + t_{11} + t_{12} + t_{13}

N*k_2 = 2pi*gamma_2 + t_{21} + t_{22} + t_{23}