If `x , y in R ,` then the determinant =`|(cos x , -sin x,1),(sin x, cos x,1),(cos(x+y), -sin(x+y), 0)|` lies in the interval (a)`[-sqrt(2),sqrt(2)]` (b) `[-1,1]` (c)`[-sqrt(2),1]` (d) `[-1,-sqrt(2)]`

Explanation

To solve the determinant problem, we need to evaluate the determinant given by:

\[
D = \begin{vmatrix}
\cos x & -\sin x & 1 \\
\sin x & \cos x & 1 \\
\cos(x+y) & -\sin(x+y) & 0
\end{vmatrix}
...