The minimum value of \(125 \tan ^{2} \theta+5 \cot ^{2} \theta\) is (a) 5 (b) 25 (c) 125 (d) 50

To make the calculation process easier, let's convert the expression to the terms of only one function, either tan(θ) or cot(θ). We choose to use the tan(θ) function, so we need to convert cot^2(θ) to tan(θ) terms. Using the relationship cot(θ) = 1/tan(θ), we get cot^2(θ) = (1/tan(θ))^2 = 1/tan^2(θ)

Now, we substitute the result in the given expression, 125*tan^2(θ) + 5*cot^2(θ), with the tan(θ) form of cot^2(θ): \(f(\theta) = 125\tan^2(\theta) + 5(1/\tan^2(\theta)) \)

To find the stationary points of the function, we need to compute the first derivative with respect to (θ). The derivative is: \(f'(\theta) = \frac{d}{d\theta}[125\tan^2(\theta) + 5(1/\tan^2(\theta))] \) Using the chain rule, we find the derivative of each term: Derivative of 125*tan^2(θ): \(250\tan(\theta)\sec^2(\theta)\) Derivative of 5*(1/tan^2(θ)): \(-10\sec^2(\theta)/\tan^3(\theta)\) Now, we can write the complete derivative: \(f'(\theta) = 250\tan(\theta)\sec^2(\theta) - 10\sec^2(\theta)/\tan^3(\theta) \)

To find the stationary points, we need to set the first derivative to zero and solve for θ: \(0 = 250\tan(\theta)\sec^2(\theta) - 10\sec^2(\theta)/\tan^3(\theta) \) To simplify the equation, we can multiply both sides by \(\tan^3(\theta)\) since it will not affect the zeros (except for the trivial cases): \(0 = 250\tan^4(\theta)\sec^2(\theta) - 10\sec^2(\theta) \) Next, we can factor out 10*sec^2(θ): \(0 = 10\sec^2(\theta)(25\tan^4(\theta) - 1) \) The equation will be equal to zero when either sec^2(θ) = 0 or 25tan^4(θ) - 1 = 0. The function sec^2(θ) doesn't have any zeros, so we can focus on the second part: \(25\tan^4(\theta) - 1 = 0\) Solving for tan^4(θ): \(\tan^4(\theta) = 1/25\) And, \(\tan^2(\theta) = \pm \sqrt[2]{1/25}\) Since tan^2(θ) is squared, it must be positive, so we get the following value: \(\tan^2(\theta) = \sqrt[2]{1/25}\)

Now we will substitute the value of tan^2(θ) that we found in Step 4 into the original expression: \(f(\theta) = 125\tan^2(\theta) + 5(1/\tan^2(\theta)) = 125(\sqrt[2]{1/25}) + 5(1/\sqrt[2]{1/25})\) \(f(\theta) = 5\sqrt[2]{25} + 5\sqrt[2]{25} = 10\sqrt[2]{25}\) The minimum value of the given expression is: \(f(\theta) = 10\sqrt[2]{25} = 50 \) Thus, the correct answer is (d) 50.