Value of \( \sec^{2}(\tan^{-1}3)+\mathrm{cosec}^{2}(\cot^{-1}4) \) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 27
Let \( \theta=\tan^{-1}3 \). Then \( \tan\theta=3 \), and by \( \sec^{2}\theta=1+\tan^{2}\theta \):
\( \sec^{2}\theta=1+9=10 \)
Let \( \varphi=\cot^{-1}4 \). Then \( \cot\varphi=4 \), and by \( \mathrm{cosec}^{2}\varphi=1+\cot^{2}\varphi \):
\( \mathrm{cosec}^{2}\varphi=1+16=17 \)
The value is \( 10+17=27 \).