Match the following:

Question

Professor PC Thomas Classes · Answer

Let
$
y=\frac{7+6 	an 	heta-	an ^2 	heta}{\left(1+	an ^2 	hetaight)}=7 \cos ^2 	heta+6 \sin 	heta \cos 	heta-\sin ^2 	heta
$
$=7\left(\frac{1+\cos 2 	heta}{2}ight)+3 \sin 2 	heta-\left(\frac{1-\cos 2 	heta}{2}ight)$
$=3 \sin 2 	heta+4 \cos 2 	heta+3-\sqrt{\left(3^2+4^2ight)}+3 \leq 3 \sin 2 	heta+4 \cos 2 	heta+3 \leq \sqrt{\left(3^2+4^2ight)}+3$
$\begin{aligned} & 	herefore-2 \leq y \leq 8 \Rightarrow \lambda=8, \mu=-2 \ & \Rightarrow \lambda+\mu=6, \lambda-\mu=10(r, s)\end{aligned}$