Chứng minh rằng với mọi \(\alpha \), ta luôn có
a) \(\sin (\alpha + {\pi \over 2}) = \cos \alpha \);
b) \({\rm{cos}}(\alpha + {\pi \over 2}) = - \sin \alpha \);
c) \(\tan (\alpha + {\pi \over 2}) = - \cot \alpha \);
d) \(\cot (\alpha + {\pi \over 2}) = - \tan \alpha \).
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Gợi ý làm bài
a) \(\sin (\alpha + {\pi \over 2}) = \sin ({\pi \over 2} - ( - \alpha )) = c{\rm{os( - }}\alpha {\rm{) = cos}}\alpha \)
b) \({\rm{cos}}(\alpha + {\pi \over 2}) = c{\rm{os(}}{\pi \over 2} - ( - \alpha ) = \sin ( - \alpha ) = - \sin \alpha \)
c) \(\tan (\alpha + {\pi \over 2}) = {{\sin (\alpha + {\pi \over 2})} \over {\cos (\alpha + {\pi \over 2})}} = {{\cos \alpha } \over { - \sin \alpha }} = - \cot \alpha \)
d) \(\cot (\alpha + {\pi \over 2}) = {{\cos (\alpha + {\pi \over 2})} \over {\sin (\alpha + {\pi \over 2})}} = {{ - \sin \alpha } \over {\cos \alpha }} = - \tan \alpha \)