The value of

Question:

The value of $\tan ^{2}\left(\sec ^{-1} 3\right)+\cot ^{2}\left(\operatorname{cosec}^{-1} 4\right)$ is _________________.

Solution:

$\tan ^{2}\left(\sec ^{-1} 3\right)+\cot ^{2}\left(\operatorname{cosec}^{-1} 4\right)$

$=\sec ^{2}\left(\sec ^{-1} 3\right)-1+\operatorname{cosec}^{2}\left(\operatorname{cosec}^{-1} 4\right)-1$                               $\left(1+\tan ^{2} \theta=\sec ^{2} \theta\right.$ and $\left.1+\cot ^{2} \theta=\operatorname{cosec}^{2} \theta\right)$

$=\left[\sec \left(\sec ^{-1} 3\right)\right]^{2}+\left[\operatorname{cosec}\left(\operatorname{cosec}^{-1} 4\right)\right]^{2}-2$

$=3^{2}+4^{2}-2$

$=9+16-2$

$=23$

Thus, the value of $\tan ^{2}\left(\sec ^{-1} 3\right)+\cot ^{2}\left(\operatorname{cosec}^{-1} 4\right)$ is 23 .

The value of $\tan ^{2}\left(\sec ^{-1} 3\right)+\cot ^{2}\left(\operatorname{cosec}^{-1} 4\right)$ is ___23____.

 

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