Question

Prove each identity. (All identities in this chapter can be proven. ) $$\frac{\csc \theta}{\sec \theta}=\cot \theta$$

Short answer

The identity \(\frac{\csc\theta}{\sec\theta}=\cot\theta\) is proven by expressing the left side in terms of sine and cosine, which simplifies to \(\cot\theta\).

Step-by-step solution

Express the left side in terms of sine and cosine

Rewrite the left side of the equation using the definitions of cosecant and secant in terms of sine and cosine. We know that \(\csc\theta = \frac{1}{\sin\theta}\) and \(\sec\theta = \frac{1}{\cos\theta}\). Thus, the left side of the identity becomes \(\frac{\frac{1}{\sin\theta}}{\frac{1}{\cos\theta}}\).

Multiply by the reciprocal

Instead of dividing by a fraction, multiply by its reciprocal to simplify the expression. The left side becomes \(\frac{1}{\sin\theta} \times \frac{\cos\theta}{1}\) which simplifies to \(\frac{\cos\theta}{\sin\theta}\).

Recognize the definition of cotangent

Recognize that \(\cot\theta\) is defined as \(\frac{\cos\theta}{\sin\theta}\). Therefore, the left side simplifies to \(\cot\theta\), which is exactly the right side of the original identity.

Key concepts

Cosecant Definition

The cosecant function is one of the six fundamental trigonometric functions, and it's a bit less commonly used than sin, cos, and tan. At its core, the cosecant (written as \(\csc\)) is the reciprocal of the sine function. This means that for any angle \(\theta\), the cosecant is defined as:
\[\csc(\theta) = \frac{1}{\sin(\theta)}\]
Understanding this definition is crucial because it allows us to rewrite trigonometric expressions in terms of their reciprocals, leading to simplification opportunities, especially when encountered in equations or identities.

Secant Definition

The secant function is another one of the trigonometric functions. Similar to the cosecant, it plays its role as the reciprocal of a more commonly known function, the cosine. For any angle \(\theta\), the secant is symbolized as \(\sec\) and defined as:
\[\sec(\theta) = \frac{1}{\cos(\theta)}\]
By understanding that \(\sec\) represents the reciprocal of \(\cos\), we can transform expressions involving secant into their cosine counterparts, which is a strategy often used in solving trigonometric identities.

Cotangent Definition

The cotangent function, often abbreviated as \(\cot\), is the third reciprocal trigonometric function we’ll discuss. It is the reciprocal of the tangent function, with the added nuance that it can also be expressed as the ratio of the cosine to the sine for any angle \(\theta\). Its definition is as follows:
\[\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}\]
Recognizing this ratio as cotangent is a key step in proving many trigonometric identities. It is an elegant reminder of the interconnectedness of these trigonometric functions.

Simplifying Expressions

In mathematics, simplifying expressions is the process of rewriting them in the most concise and elegant form without changing their value. Trigonometric identities often involve such simplifications. For example, by applying the reciprocal relationships and performing operations like multiplication instead of division, the expressions can often be expressed more simply. The goal is to recognize various trigonometric functions and utilize their properties and definitions to break down more complicated structures into ones that are more direct and easier to work with.
When dealing with ratios of trigonometric functions, always look for ways to express them in a simpler form, potentially even ending up with another standard trigonometric function, which is precisely what occurs with some identities.

Reciprocal of a Fraction

The reciprocal of a fraction flips its numerator and denominator. For any nonzero fraction \(\frac{a}{b}\), the reciprocal would be \(\frac{b}{a}\). Understanding this concept is key in simplifying expressions, especially within the realm of trigonometry. When dividing by a fraction, one can multiply by its reciprocal instead, which is usually a simpler operation and helps to avoid common mistakes. In the context of trigonometric identities, using reciprocals can help reveal equivalences between different forms of the same expression - turning a complex operation into a more manageable one.