\[\begin{array}{l}
1)\,\,\,6\sin x - 6\cos x - \sin x\cos x = 6\\
\Leftrightarrow 6\left( {\sin x - \cos x} \right) - \sin x\cos x = 6\,\,\,\,\left( * \right)\\
Dat\,\,\,t = \sin x - \cos x\,\,\,\left( { - \sqrt 2 \le t \le \sqrt 2 } \right)\\
\Rightarrow {t^2} = 1 - 2\sin x\cos x \Rightarrow \sin x\cos x = \frac{{1 - {t^2}}}{2}\\
\Rightarrow 6t - \frac{{1 - {t^2}}}{2} = 6 \Leftrightarrow 12t - 1 + {t^2} = 12\\
\Leftrightarrow {t^2} + 12t - 13 = 0 \Leftrightarrow \left[ \begin{array}{l}
t = 1\,\,\left( {tm} \right)\\
t = - 13\,\,\,\left( {ktm} \right)
\end{array} \right.\\
\Rightarrow \sin x - \cos x = 1 \Leftrightarrow \sqrt 2 \sin \left( {x - \frac{\pi }{4}} \right) = 1\\
\Leftrightarrow \sin \left( {x - \frac{\pi }{4}} \right) = \frac{1}{{\sqrt 2 }} = \sin \frac{\pi }{4}\\
\Leftrightarrow \left[ \begin{array}{l}
x - \frac{\pi }{4} = \frac{\pi }{4} + k2\pi \\
x - \frac{\pi }{4} = \frac{{3\pi }}{4} + k2\pi
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \frac{\pi }{2} + k2\pi \\
x = \pi + k2\pi
\end{array} \right.\left( {k \in Z} \right).\\
2)\,\,S = C_{46}^1 + 7C_{46}^2 + .... + \left( {{3^{46}} - 2} \right)C_{46}^{46}\\
= C_{46}^1 + \left( {{3^2} - 2} \right)C_{46}^2 + .... + \left( {{3^{46}} - 2} \right)C_{46}^{46}\\
= C_{46}^1 + {3^2}C_{46}^2 + .... + {3^{46}}C_{46}^{46} - 2\left( {C_{46}^2 + C_{46}^3 + ..... + C_{46}^{46}} \right)\\
= - 1 + C_{46}^0 + 3C_{46}^1 + {3^2}C_{46}^2 + .... + {3^{46}}C_{46}^{46} - 2\left[ {C_{46}^0 + C_{46}^1 + C_{46}^2 + C_{46}^3 + ..... + C_{46}^{46} - \left( {C_{46}^0 + C_{46}^1} \right)} \right]\\
Ap\,\,dung\,\,\,nhi\,\,thuc:\\
{\left( {x + 3} \right)^{46}} = C_{46}^0{x^{46}} + C_{46}^1{x^{45}}.3 + {3^2}C_{46}^2{x^{44}} + .... + {3^{46}}C_{46}^{46}\\
{\left( {x + 1} \right)^{46}} = C_{46}^0{x^{46}} + C_{46}^1{x^{45}} + C_{46}^2{x^{44}} + .... + C_{46}^{46}\\
\Rightarrow S = - 1 + {\left( {1 + 3} \right)^{46}} - 2\left[ {{{\left( {1 + 1} \right)}^{46}} - \left( {C_{46}^0 + C_{46}^1} \right)} \right]\\
= - 1 + {4^{46}} - 2\left( {{2^{46}} - 47} \right) = {4^{46}} - 1 - {2.4^{46}} + 2.47\\
= 93 - {4^{46}}.
\end{array}\]
