Correct answer : (A)
∆ = \(\begin{vmatrix} cosec\,x &sec\,x & sec\,x \\[0.3em] sec\,x & cosec\,x & sec\,x \\[0.3em] sec\,x &sec\,x & cosec\,x \end{vmatrix}\)
C1 → C1 + C2 + C3
∆ = \(\begin{vmatrix} cosec\,x+2\,sec\,x &sec\,x & sec\,x \\[0.3em] 2\,sec\,x+cosec\,x & cosec\,x & sec\,x \\[0.3em] 2\,sec\,x +cosec\,x&sec\,x & cosec\,x \end{vmatrix}\)
∆ = \(\begin{vmatrix} 1 &sec\,x & sec\,x \\[0.3em] 1 & cosec\,x & sec\,x \\[0.3em] 1&sec\,x & cosec\,x \end{vmatrix}\)
∆ = (cosec x+ 2sec x )[(cosec x - sec x)2]
Case1 :
(cosecc x + 2sec x ) = 0
tan x = \(-\frac{1}{2}\)
(1st real root)
Case : 2
(cosec x - sec x)2 =0
tan x = 1
(2nd real root)