Problem 60 - Algebra and Trigonometry

Question

Verify each identity. $$ \sin ^{2} \frac{\theta}{2}=\frac{\csc \theta-\cot \theta}{2 \csc \theta} $$

Step-by-Step Solution

Answer

Both sides of the equation \(\sin^{2}(\frac{\theta}{2}) = \frac{\csc\theta - \cot\theta}{2\csc\theta}\) have been evaluated to \(\sin^2(\frac{\theta}{2})\), thereby confirming the original identity as valid.

Step 1: Convert to Sine and Cosine

The first step to simplifying this identity is to convert all trigonometric functions in terms of sine and cosine. Therefore, the right-side \(\frac{\csc \theta - \cot \theta}{2 \csc \theta}\) is written as \(\frac{1/\sin(\theta) - \cos(\theta)/\sin(\theta)}{2/\sin(\theta)}\).

Step 2: Simplify the Right-Side Expression

Expressions can often be simplified by finding common denominators. Simplify the numerator of the fraction, which gives us \(\frac{1 - \cos(\theta)}{\sin(\theta)}\) and the denominator as \(\frac{2}{\sin(\theta)}\). This simplifies to \(\frac{1 - \cos(\theta)}{2}\) after cancelling the sine denominators.

Step 3: Apply Half Angle Identity

Apply the half-angle identity \(1 - \cos(\theta) = 2\sin^2(\frac{\theta}{2})\). Thus, \(\frac{1 - \cos(\theta)}{2}\) is equal to \(\sin^2(\frac{\theta}{2})\).

Key Concepts

Sine and Cosine Functions

To tackle problems involving trigonometric identities, it is essential to have a strong understanding of sine and cosine functions. These functions are the foundation of trigonometry and are frequently used in identities and equations. The sine function, denoted as \( \sin(\theta) \), represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. On the other hand, the cosine function, denoted as \( \cos(\theta) \), is the ratio of the adjacent side to the hypotenuse.

These functions are periodic, which means their values repeat at regular intervals. The sine and cosine functions are also related through the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \). This identity is pivotal since it allows conversions between sine and cosine, thus aiding in trigonometric simplification and proving identities.

Trigonometric Simplification

Simplifying trigonometric expressions is a common task in mathematics, often necessary to verify identities or solve equations. It involves rewriting expressions in simpler or more manageable forms without changing their values.

• In the given exercise, trigonometric simplification involves converting functions like cosecant (\( \csc \theta = 1/\sin \theta \)) and cotangent (\( \cot \theta = \cos \theta/\sin \theta \)) into sine and cosine.
• This conversion helps in finding a common denominator and simplifying complex expressions.

Using such conversions, the exercise demonstrates that the expression \( \frac{\csc \theta - \cot \theta}{2 \csc \theta} \) can be simplified by cancelling out common terms. This method not only makes solving the problem easier but also highlights the interconnected nature of trigonometric functions.

Half-Angle Identities

Half-angle identities are a special group of trigonometric identities that express functions of half angles, such as \( \frac{\theta}{2} \), in terms of the original angle, \( \theta \). These identities are very useful for simplifying expressions or solving equations involving angles that are not full multiples of common gradients.

One useful half-angle identity is \( \sin^2 \left( \frac{\theta}{2} \right) = \frac{1 - \cos(\theta)}{2} \). In the exercise, this identity is integral to verifying the original equation. By substituting and equating the right-hand side with this formula, the identity simplifies to \( \sin^2 \left( \frac{\theta}{2} \right) \), thus satisfying the identity's requirement.

• Half-angle identities like the one used in this solution make it easier to handle more complex expressions by breaking them down into simpler components.

Understanding and applying such identities can simplify the process of working through trigonometric proofs and enhance problem-solving skills.