Question

Without expanding, show that the value of each of the determinants is zero: `|"cos"(x+y)-sin(x+y)cos2ysinxcosxsiny-cosxsinx-cosy|`

Answer 1

`|(cos (x+y) , -sin(x+y), cos2y),(sin x, cosx , siny),(-cosx, sinx, -cosy)|`
`|(cosxcosy - sinxsiny , -sinxcosy-cosxsiny, cos^2y - sin^2y),(sinx,cosx,siny),(-cosx,sinx,-cosy)|`
`r_1-> r_1 + siny(r_2)+ cosy(r_3)`
`= |(0,0,0),(sinx, cosx, siny),(-cosx,sinx, -cosy)|`
`=0`
hence proved