Finding the Minimum Value of the Trigonometric Expression

The question asks for the minimum value of the expression \(\frac{{{a}^{2}}}{{{\cos }^{2}}x}+\frac{{{b}^{2}}}{{{\sin }^{2}}x}\), given that \(a > 0\) and \(b > 0\).

Let the given expression be \(E\):

\(E = \frac{{{a}^{2}}}{{{\cos }^{2}}x}+\frac{{{b}^{2}}}{{{\sin }^{2}}x}\)

We can rewrite this expression using the reciprocal trigonometric identities \(\sec x = \frac{1}{\cos x}\) and \(\csc x = \frac{1}{\sin x}\):

\(E = a^2 \sec^2 x + b^2 \csc^2 x\)

Using the Pythagorean identities \(\sec^2 x = 1 + \tan^2 x\) and \(\csc^2 x = 1 + \cot^2 x\), we substitute these into the expression for E:

\(E = a^2 (1 + \tan^2 x) + b^2 (1 + \cot^2 x)\)

\(E = a^2 + a^2 \tan^2 x + b^2 + b^2 \cot^2 x\)

\(E = (a^2 + b^2) + (a^2 \tan^2 x + b^2 \cot^2 x)\)

To find the minimum value of E, we need to find the minimum value of the term \((a^2 \tan^2 x + b^2 \cot^2 x)\). Since \(a > 0\), \(b > 0\), \(\tan^2 x \ge 0\), and \(\cot^2 x \ge 0\), all terms are non-negative (assuming x is such that \(\cos x \ne 0\) and \(\sin x \ne 0\)). We can apply the AM-GM (Arithmetic Mean - Geometric Mean) inequality to the terms \(a^2 \tan^2 x\) and \(b^2 \cot^2 x\).

Applying AM-GM Inequality for Minimum Value

For any non-negative real numbers x and y, the AM-GM inequality states that \(\frac{x+y}{2} \ge \sqrt{xy}\), which implies \(x+y \ge 2\sqrt{xy}\). Equality holds when \(x=y\).

Let \(x_1 = a^2 \tan^2 x\) and \(y_1 = b^2 \cot^2 x\). Applying AM-GM:

\(a^2 \tan^2 x + b^2 \cot^2 x \ge 2 \sqrt{(a^2 \tan^2 x)(b^2 \cot^2 x)}\)

\(a^2 \tan^2 x + b^2 \cot^2 x \ge 2 \sqrt{a^2 b^2 \tan^2 x \frac{1}{\tan^2 x}}\)

Assuming \(\tan x \ne 0\), we have \(\tan^2 x \frac{1}{\tan^2 x} = 1\). Since \(a > 0\) and \(b > 0\), \(\sqrt{a^2 b^2} = \sqrt{(ab)^2} = |ab| = ab\).

\(a^2 \tan^2 x + b^2 \cot^2 x \ge 2ab\)

The minimum value of \(a^2 \tan^2 x + b^2 \cot^2 x\) is \(2ab\). This minimum occurs when \(a^2 \tan^2 x = b^2 \cot^2 x\), i.e., \(a^2 \tan^2 x = \frac{b^2}{\tan^2 x}\), which leads to \(\tan^4 x = \frac{b^2}{a^2}\). Taking the square root, \(\tan^2 x = \frac{b}{a}\). Since \(a > 0\) and \(b > 0\), \(\frac{b}{a} > 0\), so there exists a real angle x satisfying this condition.

Substituting the minimum value of \(a^2 \tan^2 x + b^2 \cot^2 x\) back into the expression for E:

\(E_{min} = (a^2 + b^2) + (a^2 \tan^2 x + b^2 \cot^2 x)_{min}\)

\(E_{min} = a^2 + b^2 + 2ab\)

\(E_{min} = (a+b)^2\)

Alternative Method using Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality states that for real numbers \(u_1, u_2, v_1, v_2\), \((u_1 v_1 + u_2 v_2)^2 \le (u_1^2 + u_2^2)(v_1^2 + v_2^2)\).

Consider \(u_1 = \frac{a}{\cos x}\), \(u_2 = \frac{b}{\sin x}\), \(v_1 = \cos x\), \(v_2 = \sin x\).

Then \(u_1 v_1 + u_2 v_2 = \frac{a}{\cos x} \cos x + \frac{b}{\sin x} \sin x = a + b\).

\(u_1^2 + u_2^2 = (\frac{a}{\cos x})^2 + (\frac{b}{\sin x})^2 = \frac{a^2}{\cos^2 x} + \frac{b^2}{\sin^2 x}\).

\(v_1^2 + v_2^2 = \cos^2 x + \sin^2 x = 1\).

Applying Cauchy-Schwarz inequality:

\((a + b)^2 \le \left(\frac{a^2}{\cos^2 x} + \frac{b^2}{\sin^2 x}\right) (1)\)

\((a+b)^2 \le \frac{a^2}{\cos^2 x} + \frac{b^2}{\sin^2 x}\)

This inequality shows that the expression is always greater than or equal to \((a+b)^2\). The minimum value is \((a+b)^2\), which is attainable when \(\frac{u_1}{v_1} = \frac{u_2}{v_2}\), i.e., \(\frac{a/\cos x}{\cos x} = \frac{b/\sin x}{\sin x}\), or \(\frac{a}{\cos^2 x} = \frac{b}{\sin^2 x}\). This condition is consistent with the AM-GM approach and shows the minimum is indeed \((a+b)^2\).

Both methods confirm that the minimum value of the expression \(\frac{{{a}^{2}}}{{{\cos }^{2}}x}+\frac{{{b}^{2}}}{{{\sin }^{2}}x}\) is \((a+b)^2\).

Revision Table: Key Concepts for Minimum Value Problems

Additional Information on Trigonometric Optimization

Optimization problems involving trigonometric functions often require using identities to transform the expression. Common identities like \(\sin^2 x + \cos^2 x = 1\), \(\tan x = \frac{\sin x}{\cos x}\), \(\cot x = \frac{\cos x}{\sin x}\), \(\sec x = \frac{1}{\cos x}\), and \(\csc x = \frac{1}{\sin x}\) are fundamental.

Expressions of the form \(\frac{a^2}{\cos^2 x} + \frac{b^2}{\sin^2 x}\) or \(a^2 \sec^2 x + b^2 \csc^2 x\) are classic examples where AM-GM or Cauchy-Schwarz inequality can be effectively applied after using the \(\sec^2 x = 1 + \tan^2 x\) and \(\csc^2 x = 1 + \cot^2 x\) identities. The term \(a^2 \tan^2 x + b^2 \cot^2 x\) simplifies nicely under AM-GM because \(\tan^2 x \cdot \cot^2 x = 1\).

Calculus can also be used by treating the expression as a function of x, finding its derivative, and setting it to zero to find critical points. However, this often involves more complex differentiation and solving trigonometric equations compared to the inequality methods for this specific form.