Problem 64 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity. $$\cot ^{2} \frac{x}{2}=\frac{(1+\cos x)^{2}}{\sin ^{2} x}$$

Step-by-Step Solution

Answer

The equation is an identity as both sides are equal.

Step 1: Recall the Half Angle Identity for Cotangent

We begin by recalling the half-angle identity for cotangent: \( \cot \frac{x}{2} = \frac{1+\cos x}{\sin x} \). Squaring both sides gives us: \( \cot^2 \frac{x}{2} = \left(\frac{1+\cos x}{\sin x}\right)^2 \). This directly matches the right side of our original equation.

Step 2: Square the Right Side

Square the right-hand side of the claimed identity: \( \left(\frac{1+\cos x}{\sin x}\right)^2 = \frac{(1+\cos x)^2}{\sin^2 x} \). This shows that \( \cot^2 \frac{x}{2} \) is indeed equal to \( \frac{(1+\cos x)^2}{\sin^2 x} \), which demonstrates that the original equation is an identity.

Key Concepts

Half-Angle Formulas

Half-Angle Formulas are essential trigonometric identities that express trigonometric functions of half angles in terms of more familiar trigonometric functions of full angles. These formulas are particularly useful in simplifying expressions and solving trigonometric equations.

The Half-Angle Formula for cotangent, which is a less common but very handy identity, is given by:

• \( \cot \frac{x}{2} = \frac{1 + \cos x}{\sin x} \)

This formula is derived from the well-known tangent half-angle formula, and it allows us to express the cotangent of a half-angle purely in terms of sine and cosine of the full angle. Understanding these formulas not only simplifies calculations but also helps in verifying complex trigonometric identities.

These identities unravel the relationship between angles and are particularly helpful when dealing with integration and differentiation problems in calculus. They bridge the gap between simple algebraic expressions and trigonometric functions, allowing for easier manipulation and simplification of expressions.

Cotangent

Cotangent, often abbreviated as "cot," is one of the fundamental trigonometric functions. It is defined as the reciprocal of the tangent function. Simply put, for a given angle \( \theta \), the cotangent is:

• \( \cot \theta = \frac{1}{\tan \theta} \)
• Which is equivalent to \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

In practical terms, the cotangent represents the ratio of the adjacent side to the opposite side in a right-angled triangle. Understanding this concept is key to solving trigonometric identities and equations efficiently.

When considering half-angle identities, such as \( \cot \frac{x}{2} \), the function reveals its usefulness in simplifying expressions. By substituting known values and identities, cotangent aids in translation between various forms of mathematical expressions, making the process of verifying identities more straightforward.

Identity Verification

Identity Verification in trigonometry involves proving that two expressions are equivalently related to each other. The goal is to manipulate an expression using known identities until it appears in the form of the other side of the equation.

To verify the identity \( \cot ^{2} \frac{x}{2}=\frac{(1+\cos x)^{2}}{\sin ^{2} x} \), we start with the half-angle formula for cotangent and proceed with algebraic manipulation. Here's a step-by-step breakdown:

• Recall the identity \( \cot \frac{x}{2} = \frac{1+\cos x}{\sin x} \).
• Square both sides to get \( \cot^2 \frac{x}{2} = \left(\frac{1+\cos x}{\sin x}\right)^2 \).
• Observe that this is exactly \( \frac{(1+\cos x)^2}{\sin^2 x} \).

By demonstrating that both sides of the given equation are the same through systematic reduction and substitution, we confirm that our initial claim is indeed correct. Identity verification serves as a powerful tool to ensure correctness in mathematical proofs and reduces ambiguity in algebraic transformations.