Let P be the relation defined on the set of all real numbers such...

P = \left\{\left(a, b\right): \sec^{2}a - \tan^{2}b = 1\right\}$</p><p>$P = \left\{\left(a, a\right): \sec^{2}a - \tan^{2}a = 1\right\}$</p><p>its reflexive relation.</p><p>Now$P = \left\{\left(a, b\right): \sec^{2}a - \tan^{2}b = 1\right\}$<br /><br />$\sec^{2}a - \tan^{2}b = 1$<br /><br />$⇒ \sec^{2}a = 1 + \tan^{2}b$<br /><br />$⇒ \sec^{2}a = \sec^{2}b$<br /><br />$⇒ \left|seca\right| = \left|secb\right|$<br /><br />$\sec^{2}a - \tan^{2}b = 1$<br /><br />$1 + \tan^{2}a - \sec^{2}b + 1 = 1$<br /><br />$\sec^{2}b - \tan^{2}a = 1$<br /><br />Hence, it is symmetric.<br /><br />If \left|seca\right| = \left|secb\right|$ and \left|secb\right| = \left|secc\right|$then \left|seca\right| = \left|secc\right| ⇒$ transitive

So, it is an equivalence relation.