Problem 51 - Algebra 2

Question

Simplify each expression. \(\frac{\tan \theta \csc \theta}{\sec \theta}\)

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Write in Terms of Sine and Cosine

To simplify the expression \(\frac{\tan \theta \csc \theta}{\sec \theta}\), start by expressing each trigonometric function in terms of sine and cosine.- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)- \(\csc \theta = \frac{1}{\sin \theta}\)- \(\sec \theta = \frac{1}{\cos \theta}\)Substitute these into the expression:\[\frac{\frac{\sin \theta}{\cos \theta} \cdot \frac{1}{\sin \theta}}{\frac{1}{\cos \theta}}\]

Step 2: Simplify the Numerator

Simplify the numerator of the fraction from Step 1.\(\frac{\sin \theta}{\cos \theta} \cdot \frac{1}{\sin \theta} = \frac{1}{\cos \theta}\)Since the \(\sin \theta\) terms cancel each other out, the numerator simplifies to \(\frac{1}{\cos \theta}\).

Step 3: Simplify the Entire Expression

Now simplify the entire expression by dividing the simplified numerator by the denominator from Step 1.\[\frac{\frac{1}{\cos \theta}}{\frac{1}{\cos \theta}}\]Since the numerator and the denominator are the same, the expression simplifies to 1.

Key Concepts

Simplify Expressions

Simplifying expressions is all about making complex mathematical equations more understandable and easier to work with. To simplify, you systematically reduce the expression to its most basic form without changing its value. This is done through operations such as factoring, expanding, or cancelling terms.

• Look for common factors that can be divided out.
• Simplify within parentheses first, following the order of operations.
• Convert complicated fractions into simpler terms.

In trigonometry, simplifying might involve using algebraic identities or substituting equivalent expressions. This simplifies calculations and aids in solving more complex trigonometric problems. By simplifying, expressions become easier to evaluate or compare.

Sine and Cosine

Sine and cosine are fundamental trigonometrical functions, representing relationships between the angles and sides of a right triangle.

• Sine (\( heta\)) is the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Cosine (\( heta\)) is the ratio of the length of the adjacent side to the hypotenuse in a right triangle.

When we write other trigonometric functions in terms of sine and cosine, it can sometimes simplify the problem.
This is because sine and cosine are often more straightforward to manipulate algebraically compared to other trigonometric functions.
By using sine and cosine, you can also uncover patterns or identities that are helpful in further simplifying the expression, making calculations cleaner and more elegant.

Trigonometric Simplification

Trigonometric simplification is an important skill in math that involves reducing expressions featuring trigonometrical functions to their simplest form.

• Use known identities like \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), and \(\sec \theta = \frac{1}{\cos \theta}\).
• Substitute these equivalents to simplify complex expressions.
• Cancel out or combine like terms to achieve a more basic expression.

Through simplification, the expression \(\frac{\tan \theta \csc \theta}{\sec \theta}\) was reduced to 1 after substituting and cancelling terms.
When simplifying, always look for opportunities to apply trigonometric identities.
This not only makes the expression simpler but often reveals a deeper understanding of the relationships between the various trigonometric functions.