+ 2 + \sec^2 \theta + 2 + \csc^2 \theta = 5 + \sec^2 \theta + \csc^2 \theta

February 22, 2026

$+ 2 + \sec^2 \theta + 2 + \csc^2 \theta = 5 + \sec^2 \theta + \csc^2 \theta$

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Notice that \(\sec \theta = \frac{1}{\cos \theta}\) and \(\csc \theta = \frac{1}{\sin \theta}\). Thus, the terms simplify to:
\cos^2 \theta + 2 + \sec^2 \theta + \sin^2 \theta + 2 + \csc^2 \theta
Using the Pythagorean identity \(\cos^2 \theta + \sin^2 \theta = 1\), the expression becomes:
Recall the identities \(\sec^2 \theta = 1 + \tan^2 \theta\) and \(\csc^2 \theta = 1 + \cot^2 \theta\), so:
+ \sec^2 \theta + \csc^2 \theta = 7 + \tan^2 \theta + \cot^2 \theta
Using the AM-GM inequality, \(\tan^2 \theta + \cot^2 \theta \geq 2\), hence:

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