Rút gọn: a)1/sin2x+1/sin4x+1/sin8x+cot8x

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1. Đáp án:

   $\begin{array}{l}
   \dfrac{1}{{\sin 2x}} + \dfrac{1}{{\sin 4x}} + \dfrac{1}{{\sin 8x}} + \cot 8x\\
    = \dfrac{1}{{\sin 2x}} + \dfrac{1}{{\sin 4x}} + \dfrac{1}{{\sin 8x}} + \dfrac{{\cos 8x}}{{\sin 8x}}\\
    = \dfrac{1}{{\sin 2x}} + \dfrac{1}{{\sin 4x}} + \dfrac{{\cos 8x + 1}}{{\sin 8x}}\\
    = \dfrac{1}{{\sin 2x}} + \dfrac{1}{{\sin 4x}} + \dfrac{{2{{\cos }^2}4x}}{{2.\cos 4x.\sin 4x}}\\
    = \dfrac{1}{{\sin 2x}} + \dfrac{1}{{\sin 4x}} + \dfrac{{\cos 4x}}{{\sin 4x}}\\
    = \dfrac{1}{{\sin 2x}} + \dfrac{{\cos 4x + 1}}{{\sin 4x}}\\
    = \dfrac{1}{{\sin 2x}} + \dfrac{{2{{\cos }^2}2x}}{{2.\cos 2x.\sin 2x}}\\
    = \dfrac{1}{{\sin 2x}} + \dfrac{{\cos 2x}}{{\sin 2x}}\\
    = \dfrac{{2{{\cos }^2}x}}{{2.\cos x.\sin x}}\\
    = \dfrac{{\cos x}}{{\sin x}}\\
    = \cot x.
   \end{array}$

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