<s>[INST] Ratio of areas of two triangles Based on the figure below, what is the ratio of the area of triangle $CGI$ to the area of triangle $ABC$, in terms of $\theta$?

 [/INST] Mathematica isn’t needed for the simplification, which is quite straightforward:
$$\begin{align*}
\frac{\left({\sin\theta \over \cos\theta} - \sin\theta\tan\theta\right)^2 \tan\theta}{\sin\theta\big(\cos\theta+\sin\theta\tan\theta\big)}&=\frac{\left(\tan\theta-\sin\theta\tan\theta\right)^2\tan\theta}{\frac{\sin\theta}{\cos\theta}\left(\cos^2\theta+\sin^2\theta\right)}\\\\
&=\frac{\tan^3\theta(1-\sin\theta)^2}{\tan\theta}\\\\
&=\tan^2\theta(1-\sin\theta)^2\;.
\end{align*}$$
</s>