Giải câu 4 bài Ôn tập chương 6 sgk Đại số 10 trang 155
Câu 4: trang 155 sgk Đại số 10
Rút gọn biểu thức
a) \({{2\sin 2\alpha - \sin 4\alpha } \over {2\sin 2\alpha + \sin 4\alpha }}\)
b) \(\tan \alpha ({{1 + {{\cos }^2}\alpha } \over {\sin \alpha }} - \sin \alpha )\)
c) \({{\sin ({\pi \over 4} - \alpha ) + \cos ({\pi \over 4} - \alpha )} \over {\sin ({\pi \over 4} - \alpha ) - \cos ({\pi \over 4} - \alpha )}}\)
d) \({{\sin 5\alpha - \sin 3\alpha } \over {2\cos 4\alpha }}\)
a) \({{2\sin 2\alpha - \sin 4\alpha } \over {2\sin 2\alpha + \sin 4\alpha }} \)
\(= {{2\sin 2\alpha - 2\sin 2\alpha .cos2\alpha } \over {2\sin 2\alpha + 2\sin 2\alpha .cos2\alpha }}\)
\(= {{1 - \cos 2\alpha } \over {1 + \cos 2\alpha }} \)
\(= {{2{{\sin }^2}\alpha } \over {2{{\cos }^2}\alpha }} =\tan^2\alpha \)
b) \(\tan \alpha \left({{1 + {{\cos }^2}\alpha } \over {\sin \alpha }} - \sin \alpha\right ) \)
\(= {{\sin \alpha } \over {\cos \alpha }}\left({{1 + {{\cos }^2}\alpha - {{\sin }^2}\alpha } \over {\sin \alpha }}\right) \)
\(= {{\sin \alpha } \over {\cos \alpha }}.{{2{{\cos }^2}\alpha } \over {\sin \alpha }} \)
\(= 2\cos \alpha \)
c) \({{\sin ({\pi \over 4} - \alpha ) + \cos ({\pi \over 4} - \alpha )} \over {\sin ({\pi \over 4} - \alpha ) - \cos ({\pi \over 4} - \alpha )}}\)
\(= {{\tan \left({\pi \over 4} - \alpha \right) + 1} \over {\tan\left({\pi \over 4} - \alpha \right) - 1}} \)
\(= \left({{\tan {\pi \over 4} - \tan \alpha } \over {1 + \tan {\pi \over 4}.\tan \alpha }} + 1\right):\left({{\tan {\pi \over 4} - \tan \alpha } \over {1 + \tan {\pi \over 4}.\tan \alpha }} - 1\right) \)
\(= \left({{1 - \tan \alpha + 1 + \tan \alpha } \over {1 + \tan \alpha }} \right):\left({{1 - \tan \alpha - 1 - \tan \alpha } \over {1 + \tan \alpha }} \right) \)
\(= {{ - 1} \over {\tan \alpha }} = - \cot \alpha \)
d) \({{\sin 5\alpha - \sin 3\alpha } \over {2\cos 4\alpha }} \)
\(= {{2\cos {{5\alpha + 3\alpha } \over 2}\sin {{5\alpha - 3\alpha } \over 2}} \over {2\cos 4\alpha }} \)
\(= \sin \alpha \)
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