Verify that each equation is an identity.
cot² (x/2) = (1 + cos x)²/(sin² x)

Verified step by step guidance

1. Start by recalling the identity for cotangent: \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \). Therefore, \( \cot^2(\theta) = \left(\frac{\cos(\theta)}{\sin(\theta)}\right)^2 = \frac{\cos^2(\theta)}{\sin^2(\theta)} \).
2. Apply the double angle identity for cosine: \( \cos(2\theta) = 2\cos^2(\theta) - 1 \) or \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \).
3. Substitute \( \theta = \frac{x}{2} \) into the identity: \( \cos^2\left(\frac{x}{2}\right) = \frac{1 + \cos(x)}{2} \).
4. Express \( \cot^2\left(\frac{x}{2}\right) \) using the identity: \( \cot^2\left(\frac{x}{2}\right) = \frac{\cos^2\left(\frac{x}{2}\right)}{\sin^2\left(\frac{x}{2}\right)} = \frac{\frac{1 + \cos(x)}{2}}{1 - \cos^2\left(\frac{x}{2}\right)} \).
5. Recognize that \( \sin^2(x) = 1 - \cos^2(x) \) and simplify the expression to show that both sides of the equation are equal, confirming the identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides of the equation are defined. Common identities include the Pythagorean identities, reciprocal identities, and co-function identities. Understanding these identities is crucial for verifying equations and simplifying trigonometric expressions.

Cotangent Function

The cotangent function, denoted as cot(x), is the reciprocal of the tangent function, defined as cot(x) = cos(x)/sin(x). For half-angle identities, cot²(x/2) can be expressed in terms of sine and cosine functions, which is essential for manipulating and verifying the given equation. Recognizing how cotangent relates to sine and cosine is key in this context.

Half-Angle Formulas

Half-angle formulas provide relationships for trigonometric functions of half angles, such as sin(x/2) and cos(x/2). These formulas can be used to express cotangent and other trigonometric functions in terms of the original angle, facilitating the verification of identities. Applying these formulas correctly is vital for transforming the left-hand side of the equation into the right-hand side.