The value of \(\sec^2(\tan^{-1}2)+\mathrm{cosec}^2(\cot^{-1}3)\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 15
Let \(\theta = \tan^{-1}2\), so \(\tan\theta = 2\), and \(\phi = \cot^{-1}3\), so \(\cot\phi = 3\). The identities \(\sec^2\theta = 1 + \tan^2\theta\) and \(\mathrm{cosec}^2\phi = 1 + \cot^2\phi\) give
\(\sec^2(\tan^{-1}2) = 1 + 2^2 = 5\)
\(\mathrm{cosec}^2(\cot^{-1}3) = 1 + 3^2 = 10\)
So the value is 5 + 10 = 15.
Why the other options are incorrect
Option (2) 10 — Incorrect. This is the second term alone; the first term adds 5.
Option (3) 5 — Incorrect. This is the first term alone; the second term adds 10.
Option (4) 13 — Incorrect. This is \(\tan^2\theta + \cot^2\phi = 4 + 9\); the 1 in each identity is left out.