Problem 49 - Algebra 2

Question

Verify each identity. $$ \frac{\sec \theta}{\cot \theta+\tan \theta}=\sin \theta $$

Step-by-Step Solution

Answer

\[\frac{\sec\theta}{\cot\theta + \tan\theta} = \sin\theta\] is a valid identity.

Step 1: Rewrite in terms of sine and cosine

To simplify the left side of the equation, we start by changing all the trigonometric functions to sine and cosine. We have: \[ \frac{1/\cos\theta}{\cos\theta / \sin\theta + \sin\theta / \cos\theta} \]

Step 2: Simplification

Then, simplify the equation by taking the reciprocal where necessary and simplify further: \[\frac{1}{(\cos^2\theta / \sin\theta + \sin\theta)}\] Now multiply the numerator and denominator by \(\sin\theta\) which yields: \[\frac{\sin\theta}{\cos^2\theta + \sin^2\theta}\]

Step 3: Use Pythagorean identity

Lastly the identity \(\sin^2\theta + \cos^2\theta = 1\) is employed to further simplify: \[\frac{\sin\theta}{1} = \sin\theta\] Therefore, the original identity is verified.

Key Concepts

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that are crucial to understanding various identities and equations in trigonometry. These functions relate the angles of a right triangle to the ratios of its sides. The sine function, denoted as \( \sin \theta \), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. In contrast, the cosine function, expressed as \( \cos \theta \), is the ratio of the adjacent side to the hypotenuse.

When working with identities, trigonometric functions can often be rewritten in terms of sine and cosine. This conversion helps to simplify expressions, making it easier to verify equations. For instance, the secant function is the reciprocal of the cosine function, \( \sec \theta = \frac{1}{\cos \theta} \), and the cotangent is the reciprocal of the tangent function, \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).

By rewriting more complex functions in terms of sine and cosine, manipulations become straightforward, allowing us to break equations down into simpler parts that are easier to verify or simplify. This is exactly what happens in our example problem, where several functions are rewritten using these basic trigonometric terms.

Secant and Cotangent

Secant and cotangent are lesser-known trigonometric functions but are equally important. They can significantly aid in simplifying trigonometric expressions and verifying identities.

- **Secant Function (\(\sec\theta\))**: This function is defined as the reciprocal of the cosine function. In other words, \( \sec \theta = \frac{1}{\cos \theta} \). Using the secant function in identities often involves rewriting it in terms of \( \cos \theta \), which usually simplifies calculations.
- **Cotangent Function (\(\cot\theta\))**: Cotangent is the reciprocal of the tangent function, so \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \). It helps oftentimes to express cotangent with sine and cosine, which can simplify the handling of fractions in identities.

In the given solution, both secant and cotangent are essential to transforming the identity for simplification. By converting \( \sec \theta \) and \( \cot \theta \) into expressions involving sine and cosine functions, the manipulation of the equation becomes less complicated, leading towards verifying the identity more straightforwardly.

Pythagorean Identity

The Pythagorean identity is a cornerstone in trigonometry, often used to simplify or verify equations and identities. It states that for any angle \( \theta \), the relationship between sine and cosine is given by:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is derived from the Pythagorean theorem, which relates the sides of a right triangle. It holds true for any angle, making it a powerful tool when working with trigonometric equations.

In our example problem, the Pythagorean identity is the key step in the simplification process. After rewriting and simplifying the equation in terms of sine and cosine, the Pythagorean identity simplifies \( \cos^2 \theta + \sin^2 \theta \) to 1. This reduction leads directly to verifying the original identity:\[ \frac{\sin\theta}{1} = \sin\theta \].

Thanks to the Pythagorean identity, what might seem like a complex expression can quickly reduce into simpler terms, revealing the underlying relationships and confirming the equality of both sides of a trigonometric identity.