\( \frac{nt}{\cos r} + t \tan i \sin i - t \tan r \sin i - n t - \frac{t}{\cos i} + t = \frac{N \lambda}{2} \)
\( \frac{n}{\cos r} + \tan i \sin i - \tan r \sin i - n - \frac{1}{\cos i} + 1 = \frac{N \lambda}{2 t} \)
\( n \frac{\sin^2 r + \cos^2 r}{\cos r} + \tan i \sin i - \tan r \sin i - n - \frac{\sin^2 i + \cos^2 i}{\cos i} + 1 = \frac{N \lambda}{2 t} \)
\( n \tan r \sin r + n \cos r + \tan i \sin i - \tan r \sin i - n - \tan i \sin i - \cos i + 1 = \frac{N \lambda}{2 t} \)
\( n \tan r \sin r + n \cos r - \tan r \sin i - n - \cos i + 1 = \frac{N \lambda}{2 t} \)
\( n \tan r \sin r + n \cos r - \tan r \sin i - n + (1 - \cos i) = \frac{N \lambda}{2 t} \)
\( \frac{\sin i}{\sin r} \tan r \sin r + n \cos r - \tan r \sin i - n + (1 - \cos i) = \frac{N \lambda}{2 t} \)
\( \sin i \tan r + n \cos r - \tan r \sin i - n + (1 - \cos i) = \frac{N \lambda}{2 t} \)
\( n \cos r - n + (1 - \cos i) = \frac{N \lambda}{2 t} \)
\( n \cos r = n - (1 - \cos i) + \frac{N \lambda}{2 t} \)
\( n^2 \cos^2 r = \left ( n - (1 - \cos i) + \frac{N \lambda}{2 t} \right )^2 \)
\( n^2 (1 - \sin^2 r) = \left ( n - (1 - \cos i) + \frac{N \lambda}{2 t} \right )^2 \)
\( n^2 \left (1 - \frac{\sin^2 i}{n^2} \right ) = \left ( n - (1 - \cos i) + \frac{N \lambda}{2 t} \right )^2 \)
\( n^2 - \sin^2 i = \left ( n - (1 - \cos i) + \frac{N \lambda}{2 t} \right )^2 \)
\( \begin{array} \, n^2 - \sin^2 i \\ = \\ n^2 - n (1 - \cos i) + n \frac{N \lambda}{2 t} \\ + \\ -n (1 - \cos i) + (1 - \cos i )^2 - (1 - \cos i) \frac{N \lambda}{2 t} \, \\ + \\ \, n \frac{N \lambda}{2 t} - (1 - \cos i) \frac{N \lambda}{2 t} + \frac{N^2 \lambda^2}{4 t^2} \, \end{array} \)
\( - \sin^2 i = - 2n (1 - \cos i) + 2n \frac{N \lambda}{2 t} + (1 - 2 \cos i + \cos^2 i) - 2(1 - \cos i) \frac{N \lambda}{2 t} + \frac{N^2 \lambda^2}{4 t^2} \)
\( 0 = - 2n (1 - \cos i) + 2n \frac{N \lambda}{2 t} + (1 - 2 \cos i + \cos^2 i + \sin^2 i) - 2(1 - \cos i) \frac{N \lambda}{2 t} + \frac{N^2 \lambda^2}{4 t^2} \)
\( 0 = - 2n (1 - \cos i) + 2n \frac{N \lambda}{2 t} + (2 - 2 \cos i ) - 2(1 - \cos i) \frac{N \lambda}{2 t} + \frac{N^2 \lambda^2}{4 t^2} \)
\( 0 = - 2n (1 - \cos i) + n \frac{N \lambda}{t} + (2 - 2 \cos i ) - (1 - \cos i) \frac{N \lambda}{t} + \frac{N^2 \lambda^2}{4 t^2} \)
\( 0 = - n (1 - \cos i) 2t + n N \lambda + (1 - \cos i ) 2t - (1 - \cos i) N \lambda + \frac{N^2 \lambda^2}{4 t} \)
\( n (1 - \cos i) 2t - n N \lambda = 2t (1 - \cos i ) - N \lambda (1 - \cos i ) + \frac{N^2 \lambda^2}{4 t} \)
\( n [(1 - \cos i) 2t - N \lambda ] = (2t - N \lambda)(1 - \cos i) + \frac{N^2 \lambda^2}{4 t} \)
Zo - dat was een hele bevalling!
