Toán Lớp 9: cho góc alpha nhọn CMR
1 + tan^2 alpha=1/cos^2 alpha
1+ cot^2 alpha = 1/sin^2 alpha
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Toán Lớp 9: cho góc alpha nhọn CMR
1 + tan^2 alpha=1/cos^2 alpha
1+ cot^2 alpha = 1/sin^2 alpha
Comments ( 1 )
\tan x = \dfrac{{\sin x}}{{\cos x}},\,\,\,\,\cot x = \dfrac{{\cos x}}{{\sin x}}\\
{\sin ^2}x + {\cos ^2}x = 1\\
1 + {\tan ^2}\alpha = 1 + {\left( {\dfrac{{\sin \alpha }}{{\cos \alpha }}} \right)^2} = 1 + \dfrac{{{{\sin }^2}\alpha }}{{{{\cos }^2}\alpha }}\\
= \dfrac{{{{\cos }^2}\alpha + {{\sin }^2}\alpha }}{{{{\cos }^2}\alpha }} = \dfrac{1}{{{{\cos }^2}\alpha }}\\
1 + {\cot ^2}\alpha = 1 + {\left( {\dfrac{{\cos \alpha }}{{\sin \alpha }}} \right)^2} = 1 + \dfrac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }}\\
= \dfrac{{{{\sin }^2}\alpha + {{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} = \dfrac{1}{{{{\sin }^2}\alpha }}
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