Bài tập 22 trang 201 SGK Toán 10 NC
Bài tập 22 trang 201 SGK Toán 10 NC
Chứng minh các đẳng thức sau:
a) \({\cos ^4}\alpha - {\sin ^4}\alpha = 2{\cos ^2}\alpha - 1\)
b) \(1 - {\cot ^4}\alpha = \frac{2}{{{{\sin }^2}\alpha }} - \frac{1}{{{{\sin }^4}\alpha }}\left( {\sin \alpha \ne 0} \right)\)
c) \(\frac{{1 + {{\sin }^2}\alpha }}{{1 - {{\sin }^2}\alpha }} = 1 + 2{\tan ^2}\alpha \left( {\sin \alpha \ne \pm 1} \right)\)
a) Ta có
\(\begin{array}{*{20}{l}}
\begin{array}{l}
VT = {\cos ^4}\alpha - {\sin ^4}\alpha \\
= \left( {{{\cos }^2}\alpha + {{\sin }^2}\alpha } \right).\left( {{{\cos }^2}\alpha - {{\sin }^2}\alpha } \right)\\
= {\cos ^2}\alpha - {\sin ^2}\alpha
\end{array}\\
\begin{array}{l}
= {\cos ^2}\alpha - \left( {1 - {{\cos }^2}\alpha } \right)\\
= 2{\cos ^2}\alpha - 1 = VP
\end{array}
\end{array}\)
b)
\(\begin{array}{*{20}{l}}
\begin{array}{l}
VT = 1 - {\cot ^4}\alpha \\
= \left( {1 + {{\cot }^2}\alpha } \right).\left( {1 - {{\cot }^2}\alpha } \right)
\end{array}\\
\begin{array}{l}
= \frac{1}{{{{\sin }^2}\alpha }}\left( {\frac{{{{\sin }^2}\alpha - \left( {1 - {{\sin }^2}\alpha } \right)}}{{{{\sin }^2}\alpha }}} \right)\\
= \frac{{2{{\sin }^2}\alpha - 1}}{{{{\sin }^4}\alpha }} = \frac{2}{{{{\sin }^2}\alpha }} - \frac{1}{{{{\sin }^4}\alpha }} = VP
\end{array}
\end{array}\)
c)
\(\begin{array}{l}
VT = \frac{{1 + {{\sin }^2}\alpha }}{{1 - {{\sin }^2}\alpha }} = \frac{{1 + {{\sin }^2}\alpha }}{{{{\cos }^2}\alpha }}\\
= \frac{1}{{{{\cos }^2}\alpha }} + {\tan ^2}\alpha \\
= 1 + 2{\tan ^2}\alpha = VP
\end{array}\)
-- Mod Toán 10
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