The value of 2sec^2(tan^-12)+3 cosec^2(cot^-13) is:

The correct answer is Option (3) → 40

The identities \( \sec^2\theta=1+\tan^2\theta \) and \( \mathrm{cosec}^2\theta=1+\cot^2\theta \) give both terms directly.

With \( \theta=\tan^{-1}2 \), \( \tan\theta=2 \):

    \( \displaystyle \sec^2(\tan^{-1}2)=1+2^2=5 \)

With \( \theta=\cot^{-1}3 \), \( \cot\theta=3 \):

    \( \displaystyle \mathrm{cosec}^2(\cot^{-1}3)=1+3^2=10 \)

So the value is

    \( \displaystyle 2(5)+3(10)=10+30 \)

    \( \displaystyle =40 \)