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Chứng minh rằng:
a) \({{\sin \alpha – \sin \beta } \over {\cos \alpha – \cos \beta }} = – \sqrt 3 \) nếu
\(\left\{ \matrix{
\alpha + \beta = {\pi \over 3} \hfill \cr
\cos \alpha \ne \cos \beta \hfill \cr} \right.\)
b) \({{\cos \alpha – \cos 7\alpha } \over {\sin 7\alpha – sin\alpha }} = \tan 4\alpha \) (khi các biểu thức có nghĩa)
Đáp án
a)
Advertisements (Quảng cáo)
\(\eqalign{
& {{\sin \alpha – \sin \beta } \over {\cos \alpha – \cos \beta }} = {{2\cos {{\alpha + \beta } \over 2}\sin {{\alpha – \beta } \over 2}} \over { – 2\sin {{\alpha + \beta } \over 2}\sin {{\alpha – \beta } \over 2}}} \cr
& = – \cot {{\alpha + \beta } \over 2} = – \cot {\pi \over 6} = – \sqrt 3 \cr} \)
b)
\({{\cos \alpha – \cos 7\alpha } \over {\sin 7\alpha – sin\alpha }} = {{2\sin 4\alpha \sin 3\alpha } \over {2\cos 4\alpha \sin 3\alpha }} = \tan 4\alpha \)