UPSC CDS Math Trigonometric Ratios and Trigonometric Identities Questions

$\cos^2\theta = 1 - \frac{p^2+q^2}{2pq}$We know that the maximum valueof cos2θ is 1. Therefore, p and q cannot have opposite signs. Thus, p and q are the same.$\cos^2\theta = 1 - \frac{1}{2}\left(\frac{p}{q} + \frac{q}{p}\right)$If $\left(\frac{p}{q} + \frac{q}{p}\right)$ is more than 2, then $\cos^2\theta$ will be negative, and that is not possible. So, either both p, q are positive or negative. The minimum value of $\left(\frac{p}{q} + \frac{q}{p}\right)$ is 2 and this will happen when p = q.Thus, statement 1 is correct.$\tan^{2}\theta = \frac{4pq}{p^{2}+q^{2}+2pq} - 1$$= \frac{4}{\frac{p}{q} + \frac{q}{p} + 2} - 1$If $\left(\frac{p}{q} + \frac{q}{p}\right)$ is more than 2, then $\tan^2\theta$ will be negative and that is not possible. So, either both p, q are positive or negative. The minimum value of $\left(\frac{p}{q} + \frac{q}{p}\right)$ is 2 and this will happen when p = q.Thus, statement 2 is also correct.