Problem 40 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$\left(1-\cos ^{2} x\right) \csc x=\sin x$$

Step-by-Step Solution

Answer

Based on the step-by-step solution above, determine whether the equation \((1 - \cos^2x) \cdot \csc x = \sin x\) is an identity or not. Answer: The equation \((1 - \cos^2x) \cdot \csc x = \sin x\) is an identity.

Step 1: Identify Pythagorean identity

We can notice that \(1 - \cos^2x\) is equal to \(\sin^2x\), as shown in the Pythagorean identity: $$\sin^2x + \cos^2x = 1.$$Therefore, we can rewrite the left side of the equation as: $$\sin^2x \cdot \csc x.$$

Step 2: Simplify the expression

Now, we can simplify the left side of the equation by using the definition of the cosecant function. The cosecant function is the reciprocal of the sine function: $$\csc x = \frac{1}{\sin x}.$$Hence, we can rewrite the left side of the equation as: $$\sin^2x \cdot \frac{1}{\sin x}.$$

Step 3: Cancel the sine terms

We can cancel out the \(\sin x\) term: $$\frac{\sin^2x}{\sin x} = \sin x.$$ Since we have now simplified the left side of the equation to equal \(\sin x\), which is the right side of the equation, we can conclude that the given equation is an identity.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the fundamental identities in trigonometry that connects the sine and cosine functions. It's given by the formula:

• \( \sin^2x + \cos^2x = 1 \)

This identity is incredibly useful because it allows us to relate these two functions in a variety of trigonometric equations. For example, we can rearrange this to solve for \( \sin^2x \):

• \( \sin^2x = 1 - \cos^2x \)

In the original exercise, recognizing that \( 1 - \cos^2x \) could be replaced by \( \sin^2x \) was the key first step. This substitution helps to change the form of the equation, making it easier to handle and simplify the expressions involved.

Cosecant Function

The Cosecant function, denoted as \( \csc x \), is simply the reciprocal of the sine function. It is calculated as follows:

• \( \csc x = \frac{1}{\sin x} \)

This relationship implies that anytime you see \( \csc x \), you can replace it with \( \frac{1}{\sin x} \) in an equation. In this exercise, once we had \( \sin^2x \) on the left side, multiplying by \( \csc x \) translated to multiplying by \( \frac{1}{\sin x} \). This transformation is valuable because it can sometimes simplify and cancel terms in the equation, as was the case when the left hand was transformed and simplified easily.

Identity Proof

An identity proof in trigonometry shows that an equation is true for all values of the variable. In the exercise, the given equation was \( (1 - \cos^2x) \csc x = \sin x \).

The process to prove this involved verifying that the left side can be simplified to exactly match the right side using algebraic manipulation and known identities.

Here's a simplified breakdown of what we did:

• Use the Pythagorean Identity to substitute \( 1 - \cos^2x \) with \( \sin^2x \).
• Apply the definition of the Cosecant function to express \( \csc x \) as \( \frac{1}{\sin x} \).
• Perform algebraic simplification to cancel out terms (specifically \( \sin x \)) leading to \( \sin x \) on both sides.

These steps confirmed that the original equation is indeed an identity, as both sides become identical, demonstrating that the equation holds true for all values of \( x \) where the functions are defined.