Explanation

(b)
For collinearity, ${\ cos}x\widehat{\mathbf{i}} + \sin{x\mathbf{\ }\widehat{\mathbf{j}} =}\lambda(x\widehat{\mathbf{i}} + \sin{x\ \widehat{\mathbf{j}}})$
$\Rightarrow \cos{x = x}$
Let $f(x) = \cos{x - x}$
$\Rightarrow f^{'}(x) = - \sin{x - 1 < 0}$
$f(x)$ is decreasing function and for $x \geq \frac{\pi}{3},f(x) < 0$ and for $\frac{\pi}{3} < x < \frac{\pi}{6},\ f(x) > 0$.
Hence, unique solution exist.