Bài tập 5.87 trang 213 SBT Toán 11

Ta có:

\(\begin{array}{l}
d\left( {\tan x} \right) = \left( {\tan x} \right)\prime dx = \frac{1}{{{{\cos }^2}x}}dx\\
d\left( {\cot x} \right) = \left( {\cot x} \right)\prime dx = \frac{{ - 1}}{{{{\sin }^2}x}}dx\\
 \Rightarrow \frac{{d\left( {\tan x} \right)}}{{d\left( {\cot x} \right)}} = \frac{1}{{{{\cos }^2}x}}dx:\left( {\frac{{ - 1}}{{{{\sin }^2}x}}dx} \right) =  - \frac{{{{\sin }^2}x}}{{{{\cos }^2}x}} =  - {\tan ^2}x
\end{array}\)