91Ó°ÊÓ

The cotangent function (\text{cot}) is one of the basic trigonometric functions alongside sine, cosine, and tangent. Mathematically, cotangent is defined as the reciprocal of tangent: \( \text{cot} x = \frac{1}{\tan x} \). Alternatively, it can also be represented as the ratio of cosine to sine: \( \text{cot} x = \frac{\text{cos} x}{\text{sin} x} \). When simplifying trigonometric equations, remember these key relationships involving cotangent:

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

These definitions are useful in transforming and simplifying trigonometric expressions as was demonstrated in Step 1 of the solution where \( \text{cot} \frac{x}{2} \) was expressed as \( \frac{\text{cos} (\frac{x}{2})}{\text{sin} (\frac{x}{2})} \).

The tangent function (\text{tan}) is another primary trigonometric function. It is defined as the ratio of the sine to the cosine: \( \text{tan} x = \frac{\text{sin} x}{\text{cos} x} \). In the given exercise, the tangent at half angle was used: \( \text{tan} \frac{x}{2} = \frac{\text{sin} (\frac{x}{2})}{\text{cos} (\frac{x}{2})} \). When working with trigonometric identities, recognizing how to transform tangent is essential:

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

These relationships were used in Step 1 to express \( \text{tan} \frac{x}{2} \) and compare it to \( \text{cot} \frac{x}{2} \) facilitating the combination of fractions in Step 2.

Double-angle identities are crucial in simplifying trigonometric expressions involving angles that are twice as large as a given angle. The double-angle identities for sine and cosine are as follows:

• \( \text{sin} 2x = 2 \text{sin} x \text{cos} x \)
• \( \text{cos} 2x = \text{cos}^2 x - \text{sin}^2 x \)

In this exercise, these identities were pivotal. Specifically, in Step 3, the identity \( \text{cos}^2 (\frac{x}{2}) - \text{sin}^2 (\frac{x}{2}) = \text{cos} x \) was used. Likewise, Step 5 employed the double-angle identity to simplify the RHS by recognizing that \( \text{sin} 2x = 2 \text{sin} x \text{cos} x \). Understanding and using double-angle identities can significantly streamline the process of solving and proving trigonometric identities.

Simplifying trigonometric equations often involves transforming expressions to a more convenient or unified form. Approaches to simplification include:

• Using basic identities such as \( \text{sin}^2 x + \text{cos}^2 x = 1 \).
• Utilizing reciprocal relationships like \( \text{cot} x = \frac{1}{\text{tan} x} \).
• Employing double-angle identities to shift expressions involving angles like \( 2x \) to simpler forms.

In the exercise, the simplification was systematically applied:

• Step 1: Converted \( \text{cot} \frac{x}{2} \) and \( \text{tan} \frac{x}{2} \) for easier combination.
• Step 2: Unified these expressions over a common denominator.
• Step 3: Applied the identity \( \text{cos}^2 (\frac{x}{2}) - \text{sin}^2 (\frac{x}{2}) = \text{cos} x \).
• Steps 4 and 5: Simplified and compared both sides of the equation.

Recognizing patterns and employing these steps methodically will aid in mastering trigonometric simplification.