đơn giản các biểu thức sau:(1+cot anpha)*sin^3 anpha +(1+tan anpha)*cos^3 anpha 21/08/2021 Bởi Eloise đơn giản các biểu thức sau:(1+cot anpha)*sin^3 anpha +(1+tan anpha)*cos^3 anpha
Ta có $(1 + cot \alpha) \sin^3 \alpha + (1 + \tan \alpha) \cos^3 \alpha = (1 + \dfrac{\cos \alpha}{\sin \alpha}) \sin^3 \alpha + (1 + \dfrac{\sin \alpha}{\cos \alpha}) \cos^3 \alpha$ $= \sin^3 \alpha + \cos \alpha \sin^2 \alpha + \cos^3 \alpha + \sin \alpha \cos^2 \alpha$ $= (\sin^3 \alpha +\cos^3 \alpha) + (\cos \alpha \sin^2 \alpha+ \sin \alpha \cos^2 \alpha)$ $= (\sin \alpha + \cos \alpha)(\sin^2 \alpha + \cos^2 \alpha – \sin \alpha \cos \alpha) + \sin \alpha \cos \alpha (\sin \alpha + \cos \alpha)$ $= (\sin \alpha + \cos \alpha)(\sin^2 \alpha + \cos^2 \alpha)$ $= \sin \alpha + \cos \alpha$ Bình luận
Ta có
$(1 + cot \alpha) \sin^3 \alpha + (1 + \tan \alpha) \cos^3 \alpha = (1 + \dfrac{\cos \alpha}{\sin \alpha}) \sin^3 \alpha + (1 + \dfrac{\sin \alpha}{\cos \alpha}) \cos^3 \alpha$
$= \sin^3 \alpha + \cos \alpha \sin^2 \alpha + \cos^3 \alpha + \sin \alpha \cos^2 \alpha$
$= (\sin^3 \alpha +\cos^3 \alpha) + (\cos \alpha \sin^2 \alpha+ \sin \alpha \cos^2 \alpha)$
$= (\sin \alpha + \cos \alpha)(\sin^2 \alpha + \cos^2 \alpha – \sin \alpha \cos \alpha) + \sin \alpha \cos \alpha (\sin \alpha + \cos \alpha)$
$= (\sin \alpha + \cos \alpha)(\sin^2 \alpha + \cos^2 \alpha)$
$= \sin \alpha + \cos \alpha$
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