Explanation

(a)
Given, $\sin{\theta + cosec\ \theta = 2}$ …(i)
$\sin^{2}{\theta + cosec^{2}\theta + 2 = 4}$
$\Rightarrow \ \sin^{2}{\theta + cosec^{2}\theta = 2}$ …(ii)
$\therefore\ \sin^{4}{\theta + cosec^{4}\theta = 2}$ …(iii)
And $\left( \sin^{2}{\theta + cosec^{2}\theta} \right)^{3} = 2^{3}$
$\Rightarrow \ \sin^{6}{\theta + cosec^{6}\theta + 3\sin^{2}{\theta cosec^{2}\theta\left( \sin^{2}{\theta + cosec^{2}\theta} \right) = 8}}$
$\Rightarrow \ \sin^{6}{+ cosec^{6}\theta + 3.2 = 8}$
$\Rightarrow \sin^{6}{\theta + cosec^{6}\theta = 2}$ …(iv)
On multiplying Eqs. (iv) and (iii), we get
$\left( \sin^{4}{\theta + cosec^{4}\theta} \right)\left( \sin^{6}{\theta + cosec^{6}\theta} \right) = 4$
$\Rightarrow \ \sin^{10}{\theta + \sin^{4}{\theta cosec^{4}\theta\left( \sin^{2}{\theta + cosec^{2}\theta} \right) + cosec^{10}\theta = 4}}$
$\Rightarrow \ \sin^{10}{\theta + cosec^{10}\theta = 4 - 2 = 2}$