The value of \(\sec^2\left(\tan^{-1}2\right) + \mathrm{cosec}^2\left(\cot^{-1}3\right)\) is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 15
Let \(\tan^{-1}2 = \theta\), so \(\tan\theta = 2\). By the identity \(\sec^2\theta = 1 + \tan^2\theta\):
\(\sec^2\left(\tan^{-1}2\right) = 1 + 2^2 = 5\)
Let \(\cot^{-1}3 = \phi\), so \(\cot\phi = 3\). By the identity \(\mathrm{cosec}^2\phi = 1 + \cot^2\phi\):
\(\mathrm{cosec}^2\left(\cot^{-1}3\right) = 1 + 3^2 = 10\)
Adding:
\(5 + 10 = 15\)