Explanation

(a)
Since, $\sin\alpha,\sin{\beta\ and\ \cos\alpha}$ are in GP, then
$\sin^{2}{\beta = \sin{\alpha\cos\alpha}}$ …(i)
Given equation is $x^{2} + 2x\cot{\beta + 1 = 0}$.
$\therefore\$Discriminant, $D = b^{2} - 4ac$
$= \left( 2\cot\beta \right)^{2} - 4 = 4(cosec^{2}\beta - 2)$
$= 4(cosec\ \alpha\sec{\alpha - 2)}$ [from Eq. (i)]
$= 4(2\ cosec\ 2\alpha - 2) \geq 0$
$\therefore$ Roots are real.