Determinant value is to be found by the expansion using a column or a row. We pick the 1st column as those expressions are complicated.
= x [ 0 - 2 (1+x)² ] - 1 [ 0 - (1+x)² Cos x ] + x² [ .... ]
= - 2 x (1+x)² + (1+x)² Cos x + x² [ ... ]
Since we want only the coefficient of x (power 1), we ignore the last term. We get :
- 2 x + (1 + 2 x ) Cos x
It depends whether we can treat Cos x as a coefficient of x. We use the MaClaurin (Taylor) series for Cos x to write it as a polynomial in x.
Cos x = 1 - x²/2 + .... We ignore the x² term, as its power is > 1. So we replace Cos x by 1.
Finally, the coefficient of x : - 2 + 2 = 0