Problem 20 - Technical Mathematics with Calculus

Question

Prove each identity. $$\frac{1-\cos x}{\sin x}=\tan \frac{x}{2}$$

Step-by-Step Solution

Answer

By expressing \(\tan(\frac{x}{2})\) in terms of sine and cosine, applying double angle formulas, and simplifying, the equation is verified.

Step 1: Write tangent in terms of sine and cosine

Begin by expressing the right side of the identity in terms of sine and cosine functions using the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). The identity we want to prove is now rewritten as \(\frac{1-\cos x}{\sin x}=\frac{\sin(\frac{x}{2})}{\cos(\frac{x}{2})}\).

Step 2: Apply the double angle formulas

Use the double angle formulas \(\sin x = 2\sin(\frac{x}{2})\cos(\frac{x}{2})\) and \(\cos x = 2\cos^2(\frac{x}{2})-1\) to relate \(\sin(\frac{x}{2})\) and \(\cos(\frac{x}{2})\) to \(\sin x\) and \(\cos x\).

Step 3: Manipulate the right side of the original identity

Express the trigonometric functions of \(x\) in terms of trigonometric functions of \(\frac{x}{2}\). We have \(\sin x = 2\sin(\frac{x}{2})\cos(\frac{x}{2})\), so we can rewrite \(1-\cos x\) as\(1-(2\cos^2(\frac{x}{2})-1) = 2(1-\cos^2(\frac{x}{2}))\), which simplifies to \(2\sin^2(\frac{x}{2})\), using the Pythagorean identity \(\sin^2 \theta = 1 - \cos^2 \theta\).

Step 4: Rewrite the left side of the equation

Continue manipulating the left side of the identity: Replace \(1-\cos x\) with \(2\sin^2(\frac{x}{2})\) and \(\sin x\) with \(2\sin(\frac{x}{2})\cos(\frac{x}{2})\) to obtain \(\frac{2\sin^2(\frac{x}{2})}{2\sin(\frac{x}{2})\cos(\frac{x}{2})}\).

Step 5: Simplify the left side

Simplify the expression obtained in Step 4: \(\frac{2\sin^2(\frac{x}{2})}{2\sin(\frac{x}{2})\cos(\frac{x}{2})} = \frac{\sin(\frac{x}{2})}{\cos(\frac{x}{2})}\). After canceling a \(\sin(\frac{x}{2})\) from the numerator and denominator, we get \(\tan(\frac{x}{2})\), which matches the right side of the identity.

Key Concepts

Double Angle Formulas

Understanding double angle formulas is crucial for simplifying complex trigonometric expressions and solving equations. These identities are used to express functions of double angles, such as \( 2\theta \) in terms of single angles like \( \theta \) itself. They result from the sum formulas of trigonometric functions.

For sine, the double angle formula is given as \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \). In the context of the exercise, the double angle formula allowed us to relate \( \sin x \) to \( \sin(\frac{x}{2}) \) and \( \cos(\frac{x}{2}) \), which was a pivotal step in proving the identity.

Similarly, the double angle formula for cosine, \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \), which can also be written as \( \cos(2\theta) = 2\cos^2(\theta) - 1 \), facilitated the transformation of \( 1-\cos x \) in our original problem. If you're looking to master trigonometric concepts, a solid understanding of double angle formulas is an invaluable asset.

Pythagorean Identity

The Pythagorean identity is another cornerstone of trigonometry. It connects sine and cosine of a single angle in a beautifully simple equation: \( \sin^2\theta + \cos^2\theta = 1 \). From this fundamental relationship, we can derive other useful identities, such as \( \sin^2\theta = 1 - \cos^2\theta \) and \( \cos^2\theta = 1 - \sin^2\theta \).

In our exercise, the Pythagorean identity allowed us to rewrite \( 1-\cos x \) as \( 2\sin^2(\frac{x}{2}) \) by recognizing that \( 1-\cos^2(\theta) \) is equivalent to \( \sin^2(\theta) \). This identity is essential when simplifying expressions and proving more complex trigonometric identities. Remember, whenever you see a sum of squares for sine and cosine, the Pythagorean identity should come to mind as a potential simplification strategy.

Tangent Function

The tangent function often appears in trigonometry problems, including our example. Understanding that tangent combines both sine and cosine is the key to unlocking its properties. The basic definition of tangent in terms of sine and cosine is \( \tan\theta = \frac{\sin\theta}{\cos\theta} \).

In our exercise, expressing \( \tan(\frac{x}{2}) \) as a ratio of \( \sin(\frac{x}{2}) \) and \( \cos(\frac{x}{2}) \) was the first step to illustrate how the right side of the equation correlates to the left. This reformulation was crucial for comparing it to \( \frac{1-\cos x}{\sin x} \), leading us towards the proof of our trigonometric identity.

When working with tangent, always be mindful of the angle's quadrant and the corresponding signs of sine and cosine, as well as the potential for undefined values when \( \cos\theta = 0 \). The tangent function's unique properties provide a different perspective on trigonometry, making it indispensable for solving a wide range of problems.