Toán Lớp 9: CMR đẳng thức:
a) sin ($sin^{4}$ x – $cos^{4}$x)/ sin x+ cosx = sin x- cos x
b) $sin^{6}$ x+$cos^{6}$ x+3.$sin^{2}$x .$cos^{2}$x =1
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Toán Lớp 9: CMR đẳng thức:
a) sin ($sin^{4}$ x – $cos^{4}$x)/ sin x+ cosx = sin x- cos x
b) $sin^{6}$ x+$cos^{6}$ x+3.$sin^{2}$x .$cos^{2}$x =1
Comments ( 1 )
a,\\
\dfrac{{{{\sin }^4}x – {{\cos }^4}x}}{{\sin x + \cos x}}\\
= \dfrac{{{{\left( {{{\sin }^2}x} \right)}^2} – {{\left( {{{\cos }^2}x} \right)}^2}}}{{\sin x + \cos x}}\\
= \dfrac{{\left( {{{\sin }^2}x – {{\cos }^2}x} \right).\left( {{{\sin }^2}x + {{\cos }^2}x} \right)}}{{\sin x + \cos x}}\\
= \dfrac{{\left( {{{\sin }^2}x – {{\cos }^2}x} \right).1}}{{\sin x + \cos x}}\\
= \dfrac{{{{\sin }^2}x – {{\cos }^2}x}}{{\sin x + \cos x}}\\
= \dfrac{{\left( {\sin x – \cos x} \right)\left( {\sin x + \cos x} \right)}}{{\sin x + \cos x}}\\
= \sin x – \cos x\\
b,\\
{\sin ^6}x + {\cos ^6}x + 3{\sin ^2}x.\cos {x^2}\\
= {\sin ^6}x + {\cos ^6}x + 3{\sin ^2}x.\cos {x^2}.1\\
= {\sin ^6}x + {\cos ^6}x + 3{\sin ^2}x.\cos {x^2}.\left( {{{\sin }^2}x + {{\cos }^2}x} \right)\\
= {\sin ^6}x + {\cos ^6}x + 3{\sin ^4}x.\cos {x^2} + 3{\sin ^2}x.{\cos ^4}x\\
= {\sin ^6}x + 3{\sin ^4}x.{\cos ^2}x + 3{\sin ^2}x.{\cos ^4}x + {\cos ^6}x\\
= {\left( {{{\sin }^2}x} \right)^3} + 3.{\left( {{{\sin }^2}x} \right)^2}.{\cos ^2}x + 3.{\sin ^2}x.{\left( {{{\cos }^2}x} \right)^2} + {\left( {{{\cos }^2}x} \right)^3}\\
= {\left( {{{\sin }^2}x + {{\cos }^2}x} \right)^3}\\
= {1^3}\\
= 1
\end{array}\)