Explanation

(b)
We Know, $\left( \sigma_{x} - \sigma_{y} \right)^{2} \geq 0$
$\Rightarrow \sigma_{x}^{2} + \sigma_{x}^{2} \geq 2\sigma_{x}\sigma_{y}$
$\Rightarrow \frac{\sigma_{x}\sigma_{y}}{\sigma_{x}^{2} + \sigma_{y}^{2}} \leq \frac{1}{2}$
If$\theta\ is\ angle\ between\ two\ regression\ lines\ with\ correlation\ coefficient\ y,\ then$ tan $\theta = \left( \frac{1 - y^{2}}{y} \right)\left( \frac{\sigma_{x}\sigma_{y}}{\sigma_{x}^{2} + \sigma_{y}^{2}} \right)$
$\Rightarrow tan\theta\ \leq \frac{1 - y^{2}}{2y}$
$\Rightarrow \tan^{2}{\theta \leq \left( \frac{1 - y^{2}}{2y} \right)^{2}}$
Since, $\sin^{2}{\theta \leq 1}\ and1 - y^{2} < 1 + y^{2}$
$\therefore sin\theta \leq 1 - y^{2}$