Answer:
Option C
Explanation:
Here, x2−2xsecθ+1=0 has roots α1 and β1.
∴ α1,β1=2secθ±√4sec2θ−42×1
=2secθ±2|tanθ|2
Since, θϵ(−π6,−π12),
i.e, Θ ε IV quadrant = 2secθ±2tanθ2
∴ α1=secθ−tanθ and β1=secθ+tanθ
(as α1 > β1 )
and x2+2xtanθ−1=0 has roots α2 and β2.
i.e. α2,β2=−2tanθ±√4tan2θ+42
∴ α2=−tanθ+secθ
and β2=−tanθ−secθ [as α2>β2 ]
Thus, α1+β2=−2tanθ