In Exercises 59–68, verify each identity.
cos²(θ/2) = (sec θ + 1)/(2 sec θ)

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Start by writing down the given identity clearly: \(\frac{\theta \sec^2 \theta}{2} + 1 = \frac{\cos^2 \theta}{2 \sec \theta}\).

Recall the definition of secant: \(\sec \theta = \frac{1}{\cos \theta}\). Use this to rewrite all secant terms in the identity in terms of cosine.

Rewrite the left-hand side (LHS) by substituting \(\sec^2 \theta\) with \(\frac{1}{\cos^2 \theta}\), so the LHS becomes \(\frac{\theta}{2 \cos^2 \theta} + 1\).

Rewrite the right-hand side (RHS) by substituting \(\sec \theta\) with \(\frac{1}{\cos \theta}\), so the RHS becomes \(\frac{\cos^2 \theta}{2 \times \frac{1}{\cos \theta}} = \frac{\cos^2 \theta \times \cos \theta}{2} = \frac{\cos^3 \theta}{2}\).

Simplify both sides as much as possible and then check if they are equal by manipulating the expressions algebraically, such as finding a common denominator or factoring, to verify the identity.

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