Rút gọn các biểu thức lượng giác sau

\(\begin{array}{l} A = \dfrac{{\cos 2x}}{{\sin x + \cos x}} \\= \dfrac{{{{\cos }^2}x - {{\sin }^2}x}}{{\sin x + \cos x}}\\ = \dfrac{{(\cos x - \sin x)(\cos x + \sin x)}}{{\sin x + \cos x}}\\ = \cos x - \sin x\\ = \sqrt 2 \cos \left( {x + \dfrac{\pi }{4}} \right) \end{array}\) \(\begin{array}{l} B = \dfrac{{\sin x + \tan x}}{{\tan x}} - \sin x.\cos x\\ = \dfrac{{\sin x + \frac{{\sin x}}{{\cos x}}}}{{\frac{{\sin x}}{{\cos x}}}} - \sin x.\cos x\\ = \dfrac{{\cos x\sin x + \sin x}}{{\cos x}}\dfrac{{\cos x}}{{\sin x}} - \sin x.\cos x\\ = \cos x + 1 - \sin x.\cos x\\ \\ C = \tan x + \dfrac{{\cos x}}{{1 + \sin x}}\\ = \dfrac{{\sin x}}{{\cos x}} + \frac{{\cos x}}{{1 + \sin x}}\\ = \dfrac{{\sin x + {{\sin }^2}x + {{\cos }^2}x}}{{\cos x(1 + \sin x)}}\\ = \dfrac{{1 + \sin x}}{{\cos x(1 + \sin x)}}\\ = \dfrac{1}{{\cos x}} \end{array}\)