Understanding the Problem: Finding the Least Value Condition for m(θ)

The problem asks us to find the condition under which the given function \(m(\theta) = \cot^2\theta + n^2\tan^2\theta + 2n\) attains its least value. Here, \(n\) is a fixed positive real number.

To find the least value of an expression, especially one involving squares of reciprocal trigonometric functions and a constant term, we often look for ways to apply inequalities like AM-GM (Arithmetic Mean - Geometric Mean).

Applying AM-GM Inequality

The expression for \(m(\theta)\) has three terms: \(\cot^2\theta\), \(n^2\tan^2\theta\), and \(2n\). The term \(2n\) is a constant since \(n\) is a fixed number. The variation in \(m(\theta)\) depends on the terms involving \(\theta\): \(\cot^2\theta + n^2\tan^2\theta\).

For \(\cot\theta\) and \(\tan\theta\) to be defined and the squares to be meaningful in the context of positive terms for AM-GM, we consider values of \(\theta\) where \(\cot\theta \neq 0\) and \(\tan\theta \neq 0\). In such cases, \(\cot^2\theta > 0\) and \(\tan^2\theta > 0\). Since \(n\) is a positive real number, \(n^2 > 0\).

Thus, \(\cot^2\theta\) and \(n^2\tan^2\theta\) are positive terms for relevant values of \(\theta\). We can apply the AM-GM inequality to these two terms:

For any two non-negative numbers \(a\) and \(b\), the AM-GM inequality states that \(\frac{a+b}{2} \ge \sqrt{ab}\), which can be rewritten as \(a+b \ge 2\sqrt{ab}\). Equality holds if and only if \(a=b\).

Let \(a = \cot^2\theta\) and \(b = n^2\tan^2\theta\). Applying the inequality:

\(\cot^2\theta + n^2\tan^2\theta \ge 2\sqrt{(\cot^2\theta)(n^2\tan^2\theta)}\)

Simplify the term under the square root:

\((\cot^2\theta)(n^2\tan^2\theta) = n^2 (\cot\theta \tan\theta)^2\)

We know that \(\tan\theta = \frac{1}{\cot\theta}\), so \(\cot\theta \tan\theta = 1\).

Thus, the term under the square root becomes:

\(n^2 (1)^2 = n^2\)

Substitute this back into the inequality:

\(\cot^2\theta + n^2\tan^2\theta \ge 2\sqrt{n^2}\)

Since \(n\) is a positive real number, \(\sqrt{n^2} = n\).

\(\cot^2\theta + n^2\tan^2\theta \ge 2n\)

Now, consider the full expression for \(m(\theta)\):

\(m(\theta) = \cot^2\theta + n^2\tan^2\theta + 2n\)

Using the inequality we just derived, we can find a lower bound for \(m(\theta)\):

\(m(\theta) \ge 2n + 2n\)

\(m(\theta) \ge 4n\)

The least value that \(m(\theta)\) can attain is \(4n\).

Condition for Attaining the Least Value

The least value is attained when the equality holds in the AM-GM inequality. This occurs when the two terms we applied AM-GM to are equal:

\(\cot^2\theta = n^2\tan^2\theta\)

We need to find the condition on \(\theta\) (or a relationship between \(n\) and a trigonometric function of \(\theta\)) that satisfies this equality. Substitute \(\tan\theta = \frac{1}{\cot\theta}\):

\(\cot^2\theta = n^2 \left(\frac{1}{\cot^2\theta}\right)\)

Multiply both sides by \(\cot^2\theta\) (assuming \(\cot^2\theta \neq 0\)):

\((\cot^2\theta)^2 = n^2\) \(\cot^4\theta = n^2\)

Taking the square root of both sides:

\(\sqrt{\cot^4\theta} = \sqrt{n^2}\) \(|\cot^2\theta| = |n|\)

Since \(\cot^2\theta\) is always non-negative and \(n\) is positive (\(n > 0\)), we have:

\(\cot^2\theta = n\)

This is the condition under which the minimum value of \(m(\theta)\) is attained. The question asks for the condition in terms of \(n\). From our derivation, the condition is \(n = \cot^2\theta\).

Comparing with Options

Let's look at the given options:

1. \(n = \tan^2\theta\)
2. \(n = \cot^2\theta\)
3. \(n = \sin^2\theta\)
4. \(n = \cos^2\theta\)

Our derived condition for the least value of \(m(\theta)\) is \(n = \cot^2\theta\).

This matches Option 2.

Conclusion

The function \(m(\theta) = \cot^2\theta + n^2\tan^2\theta + 2n\) attains its least value when the terms \(\cot^2\theta\) and \(n^2\tan^2\theta\) are equal. This equality condition simplifies to \(n = \cot^2\theta\).

Revision Table: Key Concepts

Additional Information: Understanding Minimums

When we talk about the minimum value of a function like \(m(\theta)\), we are looking for the lowest possible output value the function can produce for any valid input \(\theta\). The condition under which this minimum is achieved tells us the relationship between the variables (\(n\) and \(\theta\)) that makes this happen.

In this problem, the AM-GM inequality provides a lower bound for the expression \(\cot^2\theta + n^2\tan^2\theta\). This lower bound is \(2n\). Since the inequality can achieve equality, the minimum value of \(\cot^2\theta + n^2\tan^2\theta\) is exactly \(2n\). Consequently, the minimum value of \(m(\theta) = (\cot^2\theta + n^2\tan^2\theta) + 2n\) is \(2n + 2n = 4n\).

The condition \(n = \cot^2\theta\) is what makes the terms equal, thus achieving the minimum sum \(2n\).

It's important to note that for \(\cot^2\theta\) to be equal to a positive number \(n\), there must exist values of \(\theta\) for which this is true. Since \(\cot^2\theta\) can take any positive value (for \(\theta\) not being a multiple of \(\pi/2\)), such \(\theta\) values always exist for any positive \(n\).