Rút gọn các biểu thức

a) \(E = \frac{{1 - {{\sin }^2}\alpha {{\cos }^2}\alpha }}{{{{\cos }^2}\alpha }} - {\cos ^2}\alpha \).

b) \(F = 2\left( {{{\sin }^4}x + {{\cos }^4}x} \right) + 4{\sin ^2}x{\cos ^2}x\).

\(A = \sin \left( {\pi  + x} \right) + \cos \left( {\frac{\pi }{2} - x} \right) + \cot \left( {2\pi  - x} \right) + \tan \left( {\frac{{3\pi }}{2} - x} \right)\)\( =  - \sin x + \sin x - \cot x + \cot x = 0\). Chọn C.

a) \(\sin \left( {3x - \frac{{7\pi }}{{12}}} \right) = \sin \left( { - x + \frac{\pi }{4}} \right)\)\( \Leftrightarrow \left[ \begin{array}{l}3x - \frac{{7\pi }}{{12}} =  - x + \frac{\pi }{4} + k2\pi \\3x - \frac{{7\pi }}{{12}} = \pi  - \left( { - x + \frac{\pi }{4}} \right) + k2\pi \end{array} \right.\)\( \Leftrightarrow \left[ \begin{array}{l}x = \frac{{5\pi }}{{24}} + k\frac{\pi }{2}\\x = \frac{{2\pi }}{3} + k\pi \end{array} \right.,k \in \mathbb{Z}\).

b) \(\sin 3x - \cos \left( {\frac{{3\pi }}{4} - x} \right) = 0\)\( \Leftrightarrow \cos \left( {\frac{\pi }{2} - 3x} \right) = \cos \left( {\frac{{3\pi }}{4} - x} \right)\)\( \Leftrightarrow \left[ \begin{array}{l}\frac{\pi }{2} - 3x = \frac{{3\pi }}{4} - x + k2\pi \\\frac{\pi }{2} - 3x =  - \frac{{3\pi }}{4} + x + k2\pi \end{array} \right.\)

\( \Leftrightarrow \left[ \begin{array}{l}x =  - \frac{\pi }{8} + k\pi \\x = \frac{{5\pi }}{{16}} + k\frac{\pi }{2}\end{array} \right.,k \in \mathbb{Z}\).