f(x) = 1 + \frac{1}{\sin^2 x} + \frac{1}{\cos^2 x}
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- We begin by simplifying the given function:
- f(x) = \sin^2 x + \cos^2 x + \frac{1}{\sin^2 x} + \frac{1}{\cos^2 x}
- Using the Pythagorean identity $ \sin^2 x + \cos^2 x = 1 $, this becomes:
- Let $ a = \sin^2 x $, $ b = \cos^2 x $, so $ a + b = 1 $, and $ a, b > 0 $ in $ (0, \frac{\pi}{2}) $. Then:
- f(x) = 1 + \frac{1}{a} + \frac{1}{b}
- We now minimize $ \frac{1}{a} + \frac{1}{b} $ under the constraint $ a + b = 1 $. Using the method of Lagrange multipliers or AM-HM inequality:
