sec⁡θ=a2+b2a2−b2, where a,b,∈R, gives real values of θ if and only if

Clearly,a2+b2≥a2−b2 for all ||a|≠|b∣≠0,⇒ a2+b2a2−b2≥1 or. a2+b2a2−b2≤−1∴ sec⁡θ=a2+b2a2−b2 is meaningful.Thus, sec⁡θ=a2+b2a2−b2 gives real values of θ if and only |a|≠|b|≠0