$\begin{bmatrix}1 & -tan\theta \\tan\theta & 1 \end{bmatrix}\begin{bmatrix}1 & tan\theta \\-tan\theta & 1 \end{bmatrix}^{-1}=\begin{bmatrix}a & -b \\b & a \end{bmatrix}$
$\Rightarrow\begin{bmatrix}1 & -tan\theta \\tan \theta & 1 \end{bmatrix}\frac{1}{1+tan^2\theta} \begin{bmatrix}1 & -tan\theta \\ tan\theta & 1 \end{bmatrix}=\begin{bmatrix}a & -b \\c & d \end{bmatrix}$
$\begin{bmatrix}1-tan^2\theta & -2tan\theta \\2tan\theta & 1-tan^2\theta \end{bmatrix}\frac{1}{1+tan^2\theta}=\begin{bmatrix}a &- b \\b & a \end{bmatrix}$
$\cos^2 \theta \begin{bmatrix}1-\frac{sin^2\theta}{cos^2\theta} & \frac{-2sin\theta}{cos\theta} \\\frac{2sin\theta}{cos\theta} & 1-\frac{sin^2\theta}{cos^2\theta} \end{bmatrix}=\begin{bmatrix}a & -b \\b & a \end{bmatrix}$
$a=cos2\theta,b=sin 2\theta$
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