Rút gọn các biểu thức sau:
a) \(A = \frac{{1 - 2{{\sin }^2}x}}{{1 + \sin 2x}} - \frac{{1 - \tan x}}{{1 + \tan x}}\)
b) \(B = \frac{{\sin 4x}}{{1 + \cos 4x}} \cdot \frac{{\cos 2x}}{{1 + \cos 2x}} - \cot \left( {\frac{{3\pi }}{2} - x} \right)\);
c) \(C = 2\left( {{{\cos }^4}x - {{\sin }^4}x} \right)\sin 2x\).
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a)
\(A = \frac{{1 - 2{{\sin }^2}x}}{{1 + \sin 2x}} - \frac{{1 - \tan x}}{{1 + \tan x}} = \frac{{{{\sin }^2}x + {{\cos }^2}x - 2{{\sin }^2}x}}{{{{\sin }^2}x + {{\cos }^2}x + 2\sin x\cos x}} - \frac{{1 - \frac{{\sin x}}{{\cos x}}}}{{1 + \frac{{\sin x}}{{\cos x}}}}\\ = \frac{{{{\cos }^2}x - {{\sin }^2}x}}{{{{\left( {\sin x + \cos x} \right)}^2}}} - \frac{{\cos x - \sin x}}{{\cos x + \sin x}} = \frac{{\cos x - \sin x}}{{\sin x + \cos x}} - \frac{{\cos x - \sin x}}{{\cos x + \sin x}} = 0\)
b)
\(B = \frac{{\sin 4x}}{{1 + \cos 4x}} \cdot \frac{{\cos 2x}}{{1 + \cos 2x}} - \cot \left( {\frac{{3\pi }}{2} - x} \right) = \frac{{2\sin 2x\cos 2x}}{{2{{\cos }^2}2x}}.\frac{{\cos 2x}}{{2{{\cos }^2}x}} - \cot \left( {\frac{\pi }{2} - x} \right)\\ = \frac{{\sin 2x}}{{2{{\cos }^2}x}} - \tan x = \frac{{\sin 2x - 2\sin x\cos x}}{{2{{\cos }^2}x}} = 0\)
c)
\(C = 2\left( {{{\cos }^4}x - {{\sin }^4}x} \right)\sin 2x\\ = 2\left( {{{\cos }^2}x - {{\sin }^2}x} \right)\left( {{{\cos }^2}x + {{\sin }^2}x} \right)\sin 2x\\ = 2\cos 2x.\sin 2x = \sin 4x\)