Verify that each equation is an identity.
(sin 2x)/(2sin x) = cos² (x/2) - sin² (x/2)

Verified step by step guidance

1. Start by recalling the double-angle identity for sine: \(\sin 2x = 2 \sin x \cos x\). This will help simplify the left-hand side (LHS) of the equation.
2. Substitute \(\sin 2x\) in the LHS with \(2 \sin x \cos x\), so the LHS becomes \(\frac{2 \sin x \cos x}{2 \sin x}\).
3. Simplify the fraction by canceling common factors in numerator and denominator, which should leave you with \(\cos x\) on the LHS.
4. Next, focus on the right-hand side (RHS): \(\cos^{2} \left(\frac{x}{2}\right) - \sin^{2} \left(\frac{x}{2}\right)\). Recognize this as a cosine double-angle identity, which states \(\cos 2\theta = \cos^{2} \theta - \sin^{2} \theta\).
5. Apply the identity to the RHS by letting \(\theta = \frac{x}{2}\), so the RHS simplifies to \(\cos x\). Since both sides simplify to \(\cos x\), the equation is verified as an 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 both sides simplify to the same expression using known formulas, such as double-angle or Pythagorean identities.

Double-Angle Formulas

Double-angle formulas express trigonometric functions of twice an angle in terms of functions of the original angle. For example, sin(2x) = 2 sin x cos x and cos(2x) = cos² x - sin² x, which are essential for rewriting and simplifying expressions involving multiples of angles.

Half-Angle Formulas

Half-angle formulas relate trigonometric functions of half an angle to those of the full angle, such as cos²(x/2) = (1 + cos x)/2 and sin²(x/2) = (1 - cos x)/2. These are useful for transforming expressions involving cos²(x/2) and sin²(x/2) into forms involving cos x.