Problem 18 - Algebra 2

Question

Verify that \(\tan \theta \sin \theta \cos \theta \csc ^{2} \theta=1\) is an identity.

Step-by-Step Solution

Answer

Yes, the expression is an identity as it simplifies to 1.

Step 1: Understand the Use of Identities

To solve this problem, we need to verify if the given expression \(\tan \theta \sin \theta \cos \theta \csc ^{2} \theta = 1\) holds true for all values of \(\theta\) where the trigonometric functions are defined. We'll utilize trigonometric identities to simplify the expression and check if it equals 1.

Step 2: Simplify with Trigonometric Definitions

Start by rewriting the expression \(\tan \theta \sin \theta \cos \theta \csc ^{2} \theta\) using trigonometric definitions: we know \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\csc \theta = \frac{1}{\sin \theta}\). Substitute these into the expression and simplify.

Step 3: Substitution of Definitions

Substitute the definitions into the expression:\[\frac{\sin \theta}{\cos \theta} \cdot \sin \theta \cdot \cos \theta \cdot \frac{1}{\sin^2 \theta}\]The \(\sin \theta\) in the numerator and \(\sin^2 \theta\) in the denominator partially cancel each other out.

Step 4: Cancel and Simplify

After cancellation, the simplified expression becomes:\[\frac{\sin \theta \cdot \cos \theta}{\cos \theta \cdot \sin \theta}\]Since \(\sin \theta\) and \(\cos \theta\) in the numerator and denominator can be cancelled, it simplifies to 1.

Step 5: Conclusion

With the simplification complete and the expression reduced to 1, it verifies our original expression \(\tan \theta \sin \theta \cos \theta \csc ^{2} \theta = 1\) is an identity. This holds true for any \(\theta\) where the functions are defined and not undefined due to division by zero.

Key Concepts

Simplification

Simplification in trigonometry involves reducing complex expressions to simpler forms by using known identities and definitions. In the given expression \(\tan \theta \sin \theta \cos \theta \csc ^{2} \theta\), simplification allows us to transform it into something easier to understand and work with.

To simplify an expression involving trigonometric functions, it’s helpful to substitute equivalent expressions using basic trigonometric identities. Here's a quick review of the identities used in this problem:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• \( \csc \theta = \frac{1}{\sin \theta} \)

By substituting these identities into the expression, we can cancel out terms as needed. In this specific example, observe how \( \sin \theta \) and \( \sin^2 \theta \) partially cancel, which is a key step in reaching the final simplified form. With these cancellations, we're left with a simplified result of 1, showing that the expression holds as a trigonometric identity.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, tangent, and cosecant, are fundamental in mathematical analyses, especially in problems related to angles and right-angled triangles. Each function has its definition based on the geometry of triangles and the unit circle, which helps in expressing angles in terms of different coordinates.

In this exercise, understanding each trigonometric function involved is crucial to solving the expression. Here’s a brief recap:

• \(\sin \theta\): Represents the ratio of the opposite side over the hypotenuse in a right-angled triangle.
• \(\cos \theta\): The ratio of the adjacent side over hypotenuse.
• \(\tan \theta\): The ratio of the opposite side over the adjacent side, or \(\frac{\sin \theta}{\cos \theta}\).
• \(\csc \theta\): Reciprocal of sine, or \(\frac{1}{\sin \theta}\).

The ability to transition between these functions using identities is what allows us to simplify or manipulate expressions effectively. They serve as the foundation for verifying whether an equation or expression is always true for defined values of \(\theta\).

Verification of Identities

The verification of trigonometric identities is the process of confirming that an expression is true for all valid values of the variable involved, usually \(\theta\). This often involves transforming one side of the equation into the other using known identities and algebraic manipulation, ensuring no step violates any mathematical principles.

For our problem, we begin by rewriting the given expression using identity substitutions. This technique simplifies the equation and enables us to manipulate and reduce it to its simplest form. Key steps include:

• Substituting definitions, such as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\csc ^2 \theta = \frac{1}{\sin^2 \theta}\).
• Canceling identical terms across numerators and denominators.
• Ensuring that the simplified expression equals the value it's supposed to, here 1.

This verification confirms whether the identity holds across the domain of the functions involved. It uses logical steps to ensure that we don't assume or ignore any undefined values, such as angles where specific functions may not be defined due to division by zero.