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Chapter 6: Problem 42

Simplify each expression. $$\frac{1}{\cos x}-\frac{\sin ^{2} x}{\cos x}$$

Short Answer

Expert verified
\( \text{cos} x \)

Step by step solution

01

Rewrite the Expression

Rewrite the given expression by combining the fractions with a common denominator: \[\frac{1 - \text{sin}^2 x}{\text{cos} x}\]
02

Use Pythagorean Identity

Apply the Pythagorean identity \(\text{sin}^2 x + \text{cos}^2 x = 1\). Therefore, \(1 - \text{sin}^2 x = \text{cos}^2 x\). Substitute in the expression: \ \[\frac{\text{cos}^2 x}{\text{cos} x}\]
03

Simplify the Expression

Simplify \(\frac{\text{cos}^2 x}{\text{cos} x}\) by canceling \(\text{cos} x\) in the numerator and the denominator: \[\frac{\text{cos}^2 x}{\text{cos} x} = \text{cos} x\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
When working with algebraic or trigonometric expressions, our goal is often to simplify them.
Simplifying like this involves combining terms and reducing them to their simplest form.
Let's look at our given expression: \(\frac{1}{\text{cos} x} - \frac{\text{sin}^2 x}{\text{cos} x} \).
We can notice that both terms share a common denominator: \cos x\.
By combining these terms into a single fraction, we get \(\frac{1 - \text{sin}^2 x}{\text{cos} x} \).
Pythagorean Identity
The Pythagorean identity is extremely useful in trigonometry.
One basic form of this identity is \(\text{sin}^2 x + \text{cos}^2 x = 1\).
This can be rearranged to solve for either sine or cosine.
In our case, we need \(1 - \text{sin}^2 x\), which equals \(\text{cos}^2 x\).
By substituting \(\text{cos}^2 x\) for \(1 - \text{sin}^2 x\), our expression simplifies further to \(\frac{\text{cos}^2 x}{\text{cos} x} \).
The trigonometric identities, like the Pythagorean identity, are fundamental tools in simplifying trig expressions.
They help us transform one form of an expression into a more manageable form.
Trigonometric Functions
Trigonometric functions, such as sine (sin) and cosine (cos), are the building blocks of trigonometry.
They represent the ratios of the sides of a right triangle relative to one of its angles.
In simplifying \(\frac{\text{cos}^2 x}{\text{cos} x}\), we can cancel \(\text{cos} x\) in the numerator and the denominator.
This leaves us with the simplified expression \(\text{cos} x\).
Understanding how to manipulate and simplify these functions is key.
Often, using identities and properties of trigonometric functions simplifies complex expressions neatly.

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