Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
3,\\
B = \left( {1 - {{\cos }^2}x} \right).{\tan ^2}x + \left( {1 - {{\tan }^2}x} \right)\\
= {\tan ^2}x - {\cos ^2}x.{\tan ^2}x + 1 - {\tan ^2}x\\
= - {\cos ^2}x.\dfrac{{{{\sin }^2}x}}{{{{\cos }^2}x}} + 1\\
= - {\sin ^2}x + 1\\
= {\cos ^2}x\\
4,\\
\cot x = 2 \Leftrightarrow \dfrac{{\cos x}}{{\sin x}} = 2 \Leftrightarrow \cos x = 2\sin x\\
\tan x = \dfrac{1}{{\cot x}} = \dfrac{1}{2}\\
{\sin ^2}x + {\cos ^2}x = 1 \Leftrightarrow {\sin ^2}x + {\left( {2\sin x} \right)^2} = 1 \Leftrightarrow {\sin ^2}x = \dfrac{1}{5}\\
B = \dfrac{{3{{\sin }^2}x + 5\sin x.\cos x + 2{{\cos }^2}x}}{{4{{\tan }^2}x + 5{{\cot }^2}x}}\\
= \dfrac{{3{{\sin }^2}x + 5.\sin x.\left( {2\sin x} \right) + 2.{{\left( {2\sin x} \right)}^2}}}{{4.{{\left( {\dfrac{1}{2}} \right)}^2} + {{5.2}^2}}}\\
= \dfrac{{21{{\sin }^2}x}}{{21}}\\
= {\sin ^2}x\\
= \dfrac{1}{5}
\end{array}\)
