Rút gọn các biểu thức sau :
a) (1- sin^2 x) cot^2 x + 1- cot^2 x
b) ( tan x + cot x ) ^2 – ( tan x – cot x ) ^2
c) ( x. Sin a – y. Cos a )^2 + ( x. Cos a + y. Sin a )^2
Rút gọn các biểu thức sau :
a) (1- sin^2 x) cot^2 x + 1- cot^2 x
b) ( tan x + cot x ) ^2 – ( tan x – cot x ) ^2
c) ( x. Sin a – y. Cos a )^2 + ( x. Cos a + y. Sin a )^2
Đáp án:
\(\begin{array}{l}
a)\,\,{\sin ^2}x\\
b)\,\,4\\
c)\,\,{x^2} + {y^2}.
\end{array}\)
Giải thích các bước giải:
\(\begin{array}{l}
a)\,\,\left( {1 – {{\sin }^2}x} \right){\cot ^2}x + 1 – {\cot ^2}x\\
= {\cot ^2}x – {\sin ^2}x.{\cot ^2}x + 1 – {\cot ^2}x\\
= 1 – {\sin ^2}x.\frac{{{{\cos }^2}x}}{{{{\sin }^2}x}} = 1 – {\cos ^2}x = {\sin ^2}x.\\
b)\,\,\,{\left( {\tan x + \cot x} \right)^2} – {\left( {\tan x – \cot x} \right)^2}\\
= {\tan ^2}x + 2\tan x.\cot x + {\cot ^2}x – {\tan ^2}x + 2\tan x.\cot x – {\cot ^2}x\\
= 4\tan x\cot x = 4.\\
c)\,\,\,{\left( {x\sin a – y\cos a} \right)^2} + {\left( {x\cos a + y\sin a} \right)^2}\\
= {x^2}{\sin ^2}a – 2xy\sin a\cos a + {y^2}{\cos ^2}a + {x^2}{\cos ^2}a + 2xy\sin a\cos a + {y^2}{\sin ^2}a\\
= {x^2}\left( {{{\sin }^2}a + {{\cos }^2}a} \right) + {y^2}\left( {{{\sin }^2}a + {{\cos }^2}a} \right)\\
= {x^2} + {y^2}.
\end{array}\)