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📅 March 10, 2026 👤 scraface

Mar 10, 2026

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Solution: We are given the function
f(x) = (\sec x + \cos x)^2 + (\csc x - \sin x)^2
for $ 0 < x < \frac{\pi}{2} $. We aim to find its minimum value.
First, expand both squared terms:
Add the two expressions:
f(x) = \sec^2 x + \cos^2 x + 2 + \csc^2 x + \sin^2 x - 2
f(x) = \sec^2 x + \csc^2 x + (\cos^2 x + \sin^2 x) + 2 - 2 = \sec^2 x...
Now express in terms of sine and cosine:
f(x) = \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x} + 1 = \frac{\sin^2 x +...
Using the identity $ \sin(2x) = 2\sin x \cos x $, we have:

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