Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\tan x.\cot x = \frac{{\sin x}}{{\cos x}}.\frac{{\cos x}}{{\sin x}} = 1\\
A = {\tan ^2}x + {\cot ^2}x = \left( {{{\tan }^2}x + 2\tan x.\cot x + {{\cot }^2}x} \right) - 2\tan x.\cot x = {\left( {\tan x + \cot x} \right)^2} - 2.1 = {k^2} - 2\\
B = {\tan ^3}x + {\cot ^3}x\\
= \left( {{{\tan }^3}x + 3{{\tan }^2}x.\cot x + 3\tan x.{{\cot }^2}x + {{\cot }^3}x} \right) - \left( {3{{\tan }^2}x.\cot x + 3\tan x.{{\cot }^2}x} \right)\\
= {\left( {\tan x + \cot x} \right)^3} - 3.\tan x.\cot x\left( {\tan x + \cot x} \right)\\
= {k^3} - 3.1.k = {k^3} - 3k
\end{array}\)
