Question

Simplify. $$\csc \theta \tan \theta-\tan \theta \sin \theta$$

Short answer

\(\cos \theta\)

Step-by-step solution

Rewrite the trigonometric identities

Express \(\csc \theta\) as \(\frac{1}{\sin \theta}\) and \(\tan \theta\) as \(\frac{\sin \theta}{\cos \theta}\).

Perform the multiplication

Multiply the expressions obtained in the first step. For the first term, multiply \(\frac{1}{\sin \theta}\) with \(\frac{\sin \theta}{\cos \theta}\) to get \(\frac{1}{\cos \theta}\), which is equal to \(\sec \theta\). For the second term, \(\tan \theta \sin \theta\) simplifies directly to \(\frac{\sin^2 \theta}{\cos \theta}\).

Combine the terms

Subtract the second term from the first term. \(\sec \theta - \frac{\sin^2 \theta}{\cos \theta}\). Factor out \(\frac{1}{\cos \theta}\) from both terms to combine them into a single fraction.

Simplify the expression

After factoring out \(\frac{1}{\cos \theta}\), we get \(\frac{1 - \sin^2 \theta}{\cos \theta}\). Use the Pythagorean identity \(1 - \sin^2 \theta = \cos^2 \theta\) to further simplify the expression to \(\frac{\cos^2 \theta}{\cos \theta}\).

Cancel the common factor

Cancel out one \(\cos \theta\) in the numerator and the denominator, which simplifies it to \(\cos \theta\).

Key concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved angles. These identities are invaluable tools in simplifying trigonometric expressions and solving trigonometric equations. A deeper understanding of these identities allows students to approach complex trigonometric problems with greater ease.

Common trigonometric identities include reciprocal identities like \( \csc\theta = \frac{1}{\sin\theta} \) and \( \sec\theta = \frac{1}{\cos\theta} \), ratio identities like \( \tan\theta = \frac{\sin\theta}{\cos\theta} \) and the Pythagorean identities, which will be discussed in further detail in a separate section. In the given exercise, recognition and application of reciprocal and ratio identities were the key to transforming the terms into a more manageable form.

Pythagorean Identity

The Pythagorean identity is a fundamental aspect of trigonometry, derived from the Pythagorean theorem, which relates the sides of a right-angled triangle. The most well-known form of the Pythagorean identity is \( \sin^2\theta + \cos^2\theta = 1 \) and its variations, \( 1 - \sin^2\theta = \cos^2\theta \) and \( 1 - \cos^2\theta = \sin^2\theta \). These identities allow for the conversion between different trigonometric functions, significantly simplifying complex expressions.

During the process of simplification in the exercise, the Pythagorean identity \( 1 - \sin^2\theta = \cos^2\theta \) played a crucial role. By recognizing it could replace \( 1 - \sin^2\theta \) with \( \cos^2\theta \) in the final expression, we simplified the complex fraction to a single trigonometric function, revealing the simplicity hidden within.

Trigonometric Functions

Trigonometric functions describe relationships between the angles and sides of a triangle, more commonly a right triangle. The six main trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cotangent (\(\cot\)), secant (\(\sec\)), and cosecant (\(\csc\)).

In a practical sense, these functions provide a way to calculate the unknown parts of a triangle given some known values, also enabling us to model periodic phenomena such as waves and oscillations. In the original exercise, the student needed to transform and manipulate these functions using identities. By turning the given expression into basic functions like \(\sec\theta\) and then canceling common factors, we successfully demonstrated the versatility and interconnectivity of trigonometric functions in problem-solving.