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

Explanation

`|(cos x , -sin x,1),(sin x, cos x,1),(cos(x+y), -sin(x+y), 0)|`

`|(cos x, -sin x,1),(sin x, cos x,1),(cos x cos y - sin xsiny , -sinxcosy - cosx siny , 0)|`

`r_3 -> r_3 - cosy(r_1) + siny(r_2)`

`|(cosx, - sinx,1),(sin x, cos x,1),(0,0, siny - cosy)|`

`(sin y - cos y)[cos^2 x - (-sin^2x)]`

`= (siny - cos y)(cos^2x+sin^2 x)`

`= siny - cosy`

`= -[cosy - siny]`

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