Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
cos θ (cos θ - sec θ)

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Start by rewriting the given expression \(\cos \theta (\cos \theta - \sec \theta)\), focusing on expressing all trigonometric functions in terms of sine and cosine.

Recall that \(\sec \theta\) is the reciprocal of \(\cos \theta\), so rewrite \(\sec \theta\) as \(\frac{1}{\cos \theta}\).

Substitute \(\sec \theta\) in the expression to get \(\cos \theta \left( \cos \theta - \frac{1}{\cos \theta} \right)\).

Distribute \(\cos \theta\) across the terms inside the parentheses: \(\cos \theta \cdot \cos \theta - \cos \theta \cdot \frac{1}{\cos \theta}\).

Simplify each term: \(\cos^2 \theta - 1\), ensuring no quotients remain and all functions are in terms of \(\theta\) only.

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