If `x , y in R ,` then the determinant `=|cosx-sinx1sinxcosx1cos(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

Given,
`=|cosx-sinx1sinxcosx1cos(x+y)-sin(x+y)0|`


Expanding along R3​, we have

`Δ=(siny−cosy)(cos2x+sin2x)`

`=(siny−cosy)`

`=sqrt(2)​ [1/(sqrt(2))​ siny−1/(sqrt(2)) cosy ]`


`=>sqrt(2)[cosπ/4​siny−sinπ/4​cosy]`

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