Problem 36 - Algebra and Trigonometry

Question

Verify the identity. $$ \csc x[\csc x+\sin (-x)]=\cot ^{2} x $$

Step-by-Step Solution

Answer

The identity is verified as both sides equal \( \frac{1}{\sin^2(x)} - 1 \).

Step 1: Recall Trigonometric Identities

To solve the identity, we need to recall some fundamental trigonometric identities. We know that \( \sin(-x) = -\sin(x) \) and \( \csc(x) = \frac{1}{\sin(x)} \). Also, \( \cot(x) = \frac{\cos(x)}{\sin(x)} \), which means \( \cot^2(x) = \left(\frac{\cos(x)}{\sin(x)}\right)^2 = \frac{\cos^2(x)}{\sin^2(x)} \).

Step 2: Simplify the Left Side

Start simplifying the left side by substituting the identity for \( \sin(-x) \):\[\csc(x)[\csc(x) + \sin(-x)] = \csc(x)[\csc(x) - \sin(x)] = \csc^2(x) - \csc(x) \sin(x)\]Substituting \( \csc(x) = \frac{1}{\sin(x)} \), we get:\[\frac{1}{\sin^2(x)} - \frac{1}{\sin(x)} \cdot \sin(x)\]This simplifies to:\[\frac{1}{\sin^2(x)} - 1\]

Step 3: Simplify the Right Side

The right side of the equation is already simplified as \( \cot^2(x) = \frac{\cos^2(x)}{\sin^2(x)} \).

Step 4: Apply Pythagorean Identity

Using the Pythagorean identity, \( \sin^2(x) + \cos^2(x) = 1 \), we can express \( \cos^2(x) \) as \( 1 - \sin^2(x) \).Hence,\[\cot^2(x) = \frac{1 - \sin^2(x)}{\sin^2(x)} = \frac{1}{\sin^2(x)} - \frac{\sin^2(x)}{\sin^2(x)} = \frac{1}{\sin^2(x)} - 1\]

Step 5: Verify Both Sides Are Equal

Both sides are \( \frac{1}{\sin^2(x)} - 1 \).Thus, the identity \( \csc x[\csc x + \sin (-x)] = \cot^2 x \) is verified.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc(x) \), is one of the basic trigonometric functions. It is the reciprocal of the sine function. This means that it is calculated as \( \csc(x) = \frac{1}{\sin(x)} \).
Understanding the cosecant function is essential since it appears in various trigonometric identities and equations, much like the one in the given exercise.

• This function is undefined at certain values where the sine equals zero because it involves division by zero, which is not possible.
• Typically, the values are at integer multiples of \( \pi \), such as \( 0, \pi, 2\pi, \) and so on.

**Application in Identities**
The cosecant function is crucial when dealing with identities, as it can often simplify expressions.

• In the example given, \( \csc(x) \) is used to contrast against sine in order to derive its squared form and to evaluate the identity.
• Recognizing it as an inverse function helps in manipulating and rearranging terms efficiently.

Cotangent Function

The cotangent function, represented by \( \cot(x) \), is another important trigonometric function, calculated as the reciprocal of the tangent function: \( \cot(x) = \frac{1}{\tan(x)} \), or equivalently, \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).
This function deals with the ratio of the cosine of an angle to the sine of that angle, which can simplify many trigonometric expressions.
**Key Points to Remember**

• Domains in which the cotangent function is defined exclude angles where the sine is zero, as this would again lead to division by zero.


• Because equations involve \( \cot^2(x) \), the understanding of this function allows us to manipulate the expression into terms of sine and cosine more readily.
• The cotangent identity \( \cot^2(x) \) is equal to \( \frac{\cos^2(x)}{\sin^2(x)} \), which is an essential form used to verify identities.

Pythagorean Identity

The Pythagorean identity is a fundamental principle in trigonometry. It expresses the invariant relationship between sine and cosine over the unit circle, stating:\[\sin^2(x) + \cos^2(x) = 1.\]
This identity is pivotal in transforming and verifying trigonometric expressions. In the context of our exercise, it was used to derive expressions for \( \cos^2(x) \) using \( \sin(x) \) and \( \cos(x) \).

**Useful Variations**
From the primary identity, other forms can be derived like:

• \( \sin^2(x) = 1 - \cos^2(x) \)
• \( \cos^2(x) = 1 - \sin^2(x) \)

These variations allow the breakdown of complicated trigonometric equations, as seen in the exercise solution where \( \cos^2(x) \) was transformed allowing \( \cot^2(x) \) to be restated as \( \frac{1 - \sin^2(x)}{\sin^2(x)} \).
This clever manipulation made it possible to express \( \cot^2(x) \) in the same form as the adjusted left side, confirming the initial identity.