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

Verified step by step guidance

1. Start by writing down the given identity clearly: \(\frac{\theta \sec^2 \theta}{2} + 1 = \frac{\cos^2 \theta}{2 \sec \theta}\).
2. 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.
3. Rewrite the left-hand side (LHS) by substituting \(\sec^2 \theta\) with \(\frac{1}{\cos^2 \theta}\), so the LHS becomes \(\frac{\theta}{2 \cos^2 \theta} + 1\).
4. Rewrite the right-hand side (RHS) by substituting \(\sec \theta\) with \(\frac{1}{\cos \theta}\), so the RHS becomes \(\frac{\cos^2 \theta}{2 \times \frac{1}{\cos \theta}} = \frac{\cos^2 \theta \times \cos \theta}{2} = \frac{\cos^3 \theta}{2}\).
5. 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.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing that both sides of the equation simplify to the same expression using known identities and algebraic manipulation.

Secant and Cosine Relationship

Secant (sec θ) is the reciprocal of cosine (cos θ), defined as sec θ = 1/cos θ. Understanding this relationship is crucial for rewriting expressions and simplifying terms involving secant and cosine functions.

Algebraic Manipulation in Trigonometry

Simplifying trigonometric expressions often requires algebraic skills such as factoring, finding common denominators, and combining fractions. These techniques help transform complex expressions into simpler or equivalent forms to verify identities.