$5 \cos ^{2} \theta-3 \sin ^{2} \theta+6 \sin \theta \cos \theta=7$

$5\left(\frac{1+\cos 2 \theta}{2}\right)-3\left(\frac{1-\cos 2 \theta}{2}\right)+3 \sin 2 \theta=7$

$4 \cos 2 \theta+3 \sin 2 \theta=6$

but $4 \cos 2 \theta+3 \sin 2 \theta \leq \sqrt{4^{2}+3^{2}}=5$

$\therefore $ Solution does not exist.