a) Rút gọn biểu thức:\(A = \dfrac{{2\sin x\left( {\cos x + \cos 3x + \cos 5x} \right)}}{{\cos 3x}}\) b)
a) Rút gọn biểu thức:\(A = \dfrac{{2\sin x\left( {\cos x + \cos 3x + \cos 5x} \right)}}{{\cos 3x}}\)
b) Rút gọn \(A = \dfrac{{1 + \sin \alpha }}{{\cos \alpha }} + \dfrac{{\cos \alpha }}{{1 + \sin \alpha }}.\)
c) Chứng minh \(\dfrac{{1 + \cos \alpha + \cos 2\alpha + \cos 3\alpha }}{{\sin 3\alpha + \sin 2\alpha - \sin \alpha }} = \cot \alpha \)
a) Rút gọn biểu thức\(A = \dfrac{{2\sin x\left( {\cos x + \cos 3x + \cos 5x} \right)}}{{\cos 3x}}\)
- Điều kiện \(\cos 3x \ne 0 \Leftrightarrow 3x \ne \dfrac{\pi }{2} + k\pi \Leftrightarrow x \ne \dfrac{\pi }{6} + \dfrac{{k\pi }}{3},k \in \mathbb{Z}\)
-Ta có \(A = \dfrac{{2\sin x\left( {\cos 3x + 2.\cos \left( {\dfrac{{5x + x}}{2}} \right).\cos \left( {\dfrac{{5x - x}}{2}} \right)} \right)}}{{\cos 3x}}\)
\( \Leftrightarrow A = \dfrac{{2\sin x\left( {\cos 3x + 2.\cos 3x.\cos 2x} \right)}}{{\cos 3x}}\)
\( \Leftrightarrow A = \dfrac{{2\sin x.\cos 3x\left( {1 + 2.\cos 2x} \right)}}{{\cos 3x}}\)
\( \Leftrightarrow A = 2\sin x\left( {1 + 2\cos 2x} \right)\)
b) Rút gọn \(A = \dfrac{{1 + \sin \alpha }}{{\cos \alpha }} + \dfrac{{\cos \alpha }}{{1 + \sin \alpha }}\)
- Điều kiện \(\left\{ \begin{array}{l}\cos \alpha \ne 0\\1 + \sin \alpha \ne 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}\alpha \ne \dfrac{\pi }{2} + k\pi \\\alpha \ne - \dfrac{\pi }{2} + k2\pi \end{array} \right.,k \in \mathbb{Z}\)
- Ta có \(A = \dfrac{{{{\left( {1 + \sin \alpha } \right)}^2} + {{\cos }^2}\alpha }}{{\left( {1 + \sin \alpha } \right).\cos \alpha }}\)
\( \Leftrightarrow A = \dfrac{{{{\sin }^2}\alpha + 2\sin \alpha + 1 + {{\cos }^2}\alpha }}{{\left( {1 + \sin \alpha } \right).\cos \alpha }}\)
\( \Leftrightarrow A = \dfrac{{1 + 2\sin \alpha + 1}}{{\left( {1 + \sin \alpha } \right).\cos \alpha }}\)
\( \Leftrightarrow A = \dfrac{{2 + 2\sin \alpha }}{{\left( {1 + \sin \alpha } \right).\cos \alpha }}\)
\( \Leftrightarrow A = \dfrac{{2.\left( {1 + \sin \alpha } \right)}}{{\left( {1 + \sin \alpha } \right).\cos \alpha }}\)
\( \Leftrightarrow A = \dfrac{2}{{\cos \alpha }}\)
c) Chứng minh \(\dfrac{{1 + \cos \alpha + \cos 2\alpha + \cos 3\alpha }}{{\sin 3\alpha + \sin 2\alpha - \sin \alpha }} = \cot \alpha \)
\( \Leftrightarrow VT = \dfrac{{1 + \cos 2\alpha + 2\cos \left( {\dfrac{{3\alpha + \alpha }}{2}} \right).\cos \left( {\dfrac{{3\alpha - \alpha }}{2}} \right)}}{{\sin 2\alpha + 2\cos \left( {\dfrac{{3\alpha + \alpha }}{2}} \right).\sin \left( {\dfrac{{3\alpha - \alpha }}{2}} \right)}}\)
\( \Leftrightarrow VT = \dfrac{{1 + 2{{\cos }^2}\alpha - 1 + 2\cos 2\alpha .\cos \alpha }}{{2\sin \alpha .\cos \alpha + 2\cos 2\alpha .\sin \alpha }}\)
\( \Leftrightarrow VT = \dfrac{{2\cos \alpha \left( {\cos \alpha + \cos 2\alpha } \right)}}{{2\sin \alpha .\left( {\cos \alpha + \cos 2\alpha } \right)}}\)
\( \Leftrightarrow VT = \dfrac{{\cos \alpha }}{{\sin \alpha }} = \cot \alpha = VP\left( {dpcm} \right)\)
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