Chứng minh rằng: a)\(\dfrac{{1 + 2\sin x.\cos x}}{{\left( {1 + \tan x} \right)\left( {1 + \cot x} \right)}} = \sin
Chứng minh rằng:
a)\(\dfrac{{1 + 2\sin x.\cos x}}{{\left( {1 + \tan x} \right)\left( {1 + \cot x} \right)}} = \sin x.\cos x\)
b)\(\dfrac{{\sin 2x + \sin x}}{{1 + \cos x + \cos 2x}} = \tan x\)
a) \(\dfrac{{1 + 2\sin x.\cos x}}{{\left( {1 + \tan x} \right)\left( {1 + \cot x} \right)}} = \sin x.\cos x\)
\(\begin{array}{l} \Leftrightarrow VT = \dfrac{{1 + 2\sin x.\cos x}}{{1 + \cot x + \tan x + \tan x.\cot x}}\\ \Leftrightarrow VT = \dfrac{{1 + 2\sin x.\cos x}}{{1 + \dfrac{{\cos x}}{{\sin x}} + \dfrac{{\sin x}}{{\cos x}} + 1}}\\ \Leftrightarrow VT = \dfrac{{1 + 2\sin x.\cos x}}{{2 + \dfrac{{{{\sin }^2}x + {{\cos }^2}x}}{{\sin x.\cos x}}}}\\ \Leftrightarrow VT = \dfrac{{1 + 2\sin x.\cos x}}{{2 + \dfrac{1}{{\sin x.\cos x}}}}\\ \Leftrightarrow VT = \dfrac{{1 + 2\sin x.\cos x}}{{\dfrac{{1 + 2\sin x.\cos x}}{{\sin x.\cos x}}}}\\ \Leftrightarrow VT = \left( {1 + 2\sin x.\cos x} \right).\dfrac{{\sin x.\cos x}}{{1 + 2\sin x.\cos x}}\\ \Leftrightarrow VT = \sin x.\cos x = VP\left( {dpcm} \right)\end{array}\)
b) \(\dfrac{{\sin 2x + \sin x}}{{1 + \cos x + \cos 2x}} = \tan x\)
\(\begin{array}{l} \Leftrightarrow VT = \dfrac{{2\sin x.\cos x + \sin x}}{{1 + \cos x + 2{{\cos }^2}x - 1}}\\ \Leftrightarrow VT = \dfrac{{\sin x.\left( {2\cos x + 1} \right)}}{{\cos x\left( {2\cos x + 1} \right)}}\\ \Leftrightarrow VT = \dfrac{{\sin x}}{{\cos x}} = \tan x\left( {dpcm} \right)\end{array}\)
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