\(\sec ^{2}\left(\tan ^{-1} 3\right)+\operatorname{cosec}^{2}\left(\tan ^{-1} 5\right)=\) (a) 276 (b) \([(276) / 25]\) (c) 36 (d) 6

We know that \(\tan(\tan^{-1} x) = x\) and \(\sin(\sin^{-1} x) = x\), where x represents the angle for which we want to find the trigonometric function. Also, we know that \(\sec^2(\theta)=1+\tan^2(\theta)\) and \(\operatorname{cosec}^2(\theta)=1+\operatorname{cot}^2(\theta)\), where \(\theta\) represents the angle for which we want to find the trigonometric functions. Given the equation: \(\sec^2(\tan^{-1}(3))+\operatorname{cosec}^2(\tan^{-1}(5))\) Replace \(3\) and \(5\) by their tangent and sine values using the aforementioned identities: \[\sec^2(\tan^{-1}(3)) = 1 + \tan^2(\tan^{-1}(3)) = 1 + 3^2 = 10\] Now, we need to find the cotangent value for the second term. We know that \(\cot(\theta) = \frac{1}{\tan(\theta)}\), so we have: \[\cot(\tan^{-1}(5)) = \frac{1}{\tan(\tan^{-1}(5))} = \frac{1}{5}\] Then: \[\operatorname{cosec}^2(\tan^{-1}(5)) = 1 + \cot^2(\tan^{-1}(5)) = 1 + \left(\frac{1}{5}\right)^2 = \frac{26}{25}\]

Now that we have calculated the values for both terms, we can combine them to get the final value of the expression: \[\sec^2(\tan^{-1}(3))+\operatorname{cosec}^2(\tan^{-1}(5)) = 10 + \frac{26}{25}\] At this point, we can see that the expression does not match any of the provided answer choices. However, if we combine the fractions and simplify, we can see that it matches one of the answer choices: \[\frac{250}{25} + \frac{26}{25} = \frac{276}{25}\] So the final value of the expression is \(\frac{276}{25}\), which matches answer choice (b). Solution: \(\boxed{(b)\ \frac{276}{25}}\)