the number of distinct real roots of
` |{:( sin x,,cos x,,cos x),(cos x,,sinx ,,cos x),(cos x,,cos x,,sin x):}|=0` in the interval `=pi//4 lt x lt pi//r `is

Options

A. 0
B. 2
C. 1
D. 3

Explanation

Given, `|{:("sin"x , "cos"x, "cos"x),("cos"x, "sin"x, "cos"x),("cos"x, "cos"x, "sin"x):}| =0`
Applying `C_(1) to C_(1) + C_(2) + C_(3)`
`|{:("sin"x +2"cos"x , "cos"x, "cos"x),("sin"x + 2 "cos"x, "sin"x, "cos"x),("sin"x+2"cos"x, "cos"x, "sin"x):}|`
`= (2"cos" x +"sin" x)|{:(1 , "cos"x, "cos"x),(1, "sin"x, "cos"x),(1, "cos"x, "sin"x):}|= 0`
Applying `R_(1) to R_(2) - R_(1), R_(3) to R_(3) -R_(1)`
`rArr (2"cos" x +"sin" x)|{:(1 , " cos"x, " cos"x),(0, "sin"x-"cos"x," " 0),(0, " "0, "sin"x-"cos"x):}|= 0`
`rArr (2"cos"x + "sin"x)("sin"x - "cos"x)^(2) = 0`
`rArr 2"cos"x + "sin"x = 0 "or sin"x -"cos"x = 0`
`rArr 2 "cos" x = -"sin"x "or sin"x = "cos"x`
`rArr "cot"x = -1//2 "gives no solution in "-(pi)/(4) le x le (pi)/(4)`
`"and sin "x= "cos"x rArr "tan"x = 1 rArr x = pi//4`