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

\text{P} = \left\{\left(\text{a, b}\right) \text{:} \sec^{2} \text{a} - \tan^{2} \text{b} = 1\right\}$<br /><br />$\sec^{2} \text{a} - \tan^{2} \text{b} = 1$<br /><br />$⇒ \sec^{2} \text{a} = 1 + \tan^{2} \text{b}$<br /><br />$⇒ \sec^{2} \text{a} = \sec^{2} \text{b}$<br /><br />$⇒ \left|\text{sec a}\right| = \left|\text{sec b}\right|$<br /><br />$\sec^{2} \text{a} - \tan^{2} \text{b} = 1$<br /><br />$1 + \tan^{2} \text{a} - \sec^{2} \text{b} + 1 = 1$<br /><br />$\sec^{2} \text{b} - \tan^{2} \text{a} = 1$<br /><br />Hence, it is Symmetric.<br /><br />If \left|\text{sec a}\right| = \left|\text{sec b}\right|$ and \left|\text{sec b}\right| = \left|\text{sec c}\right|$then \left|\text{sec a}\right| = \left|\text{sec c}\right| ⇒$ transitive&

So it is an equivalence relation.
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