Question

By using properties of determinants, prove that the determinant

\(\begin{vmatrix} a & sin\,x & cos\,x \\ -sin\,x & -a & 1 \\ cos\,x & 1 & a \end{vmatrix}\) 

is independent of x.

Answer 1

Given,

Now expanding along R₁, we get

△ = -(-a + sin x) [-a² - a sin x-1- cos x] + (1 + cos x) [-a - sin x + a + a cos x] 

= -[a³ + a² sin x + a + a cos x - a² sin x - a sin² x - sin x - sin x cos x] + [a cos x - sin x- sin x cos x + a cos² x]

= -a³, which is independent of x.