show that the value of each of the following determinants is zero : |(cos(x+y),-sin(x+y),cos2y)(sin x,cos x,sin y)(-cos x,sin x,-cos y) ​

Question

Without expanding, show that the value of each of the following determinants is zero :

\(\begin{vmatrix} cos(x+y)& -sin(x+y) & cos\,2y \\[0.3em] sinx & -cos\,x& sin\,y \\[0.3em] -cos\,x& sin\,x & -cos\,y\end{vmatrix}\)

Answer 1

Let Δ = \(\begin{vmatrix} cos(x+y)& -sin(x+y) & cos\,2y \\[0.3em] sinx & -cos\,x& sin\,y \\[0.3em] -cos\,x& sin\,x & -cos\,y\end{vmatrix}\)

Multiplying R₂ with sin y and R₃ with cos y we get,

As, 

R₁ = R₂ hence determinant is zero.