G1.3 Given that $x=2+\sqrt{3}$ is a root of the equation $x^{2}-(\tan \alpha+\cot \alpha) x+1=0$.
If $C=\sin \alpha \times \cos \alpha$, find the value of $C$.
Let the other root be $y,(2+\sqrt{3}) y=$ product of roots $=1 \Rightarrow y=2-\sqrt{3}$ $\tan \alpha+\cot \alpha=$ sum of roots $=2+\sqrt{3}+2-\sqrt{3}=4$
$$
\frac{\sin \alpha}{\cos \alpha}+\frac{\cos \alpha}{\sin \alpha}=4 \Rightarrow \frac{1}{\sin \alpha \cos \alpha}=4 \Rightarrow C=\sin \alpha \times \cos \alpha=\frac{1}{4}
$$