Problem 40 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{\cos x}{\sec x}+\frac{\sin x}{\csc x}=1$$

Step-by-Step Solution

Answer

The identity is verified as true using the Pythagorean Identity: \(\cos^2 x + \sin^2 x = 1\).

Step 1: Simplify the First Fraction

The identity involves the fraction \(\frac{\cos x}{\sec x}\). Recall that \(\sec x = \frac{1}{\cos x}\). Hence, \(\frac{\cos x}{\sec x}\) simplifies to \(\frac{\cos x}{\frac{1}{\cos x}} = \cos^2 x\).

Step 2: Simplify the Second Fraction

Now, consider the fraction \(\frac{\sin x}{\csc x}\). Since \(\csc x = \frac{1}{\sin x}\), \(\frac{\sin x}{\csc x}\) simplifies to \(\frac{\sin x}{\frac{1}{\sin x}} = \sin^2 x\).

Step 3: Add the Simplified Expressions

We have simplified the identity to \(\cos^2 x + \sin^2 x\). Recall the Pythagorean Identity: \(\cos^2 x + \sin^2 x = 1\). Thus, both fractions add up to 1.

Step 4: Verify the Identity

At the end of our simplification, we have shown that \(\cos^2 x + \sin^2 x = 1\) holds true according to the Pythagorean Identity, confirming that the original expression \(\frac{\cos x}{\sec x}+\frac{\sin x}{\csc x}=1\) is indeed an identity.

Key Concepts

Pythagorean identity

The Pythagorean Identity is a fundamental concept in trigonometry, often used to simplify expressions and verify identities. It originates from the Pythagorean Theorem and relates the squares of the sine and cosine functions. Specifically, the identity states that:

• \( \cos^2 x + \sin^2 x = 1 \)

Understanding this identity is crucial because it provides a direct relationship between sine and cosine, allowing us to manipulate and simplify trigonometric expressions. In our exercise, this identity is the key to verifying the given identity \( \frac{\cos x}{\sec x}+\frac{\sin x}{\csc x}=1 \). By expressing the fractions in terms of squared sine and cosine, we reveal the Pythagorean relationship. It's a versatile tool in both theoretical and applied mathematics, often simplifying complex problems into more manageable forms.

simplifying trigonometric expressions

Simplifying trigonometric expressions involves rewriting them in a simpler or more convenient form. A good strategy is to express all trigonometric functions in terms of sine and cosine, because they are the basic functions from which others are derived. In our exercise, the goal is to simplify:

• \( \frac{\cos x}{\sec x} \)
• \( \frac{\sin x}{\csc x} \)

To simplify these, recall that \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \). By substituting these definitions into the expressions, we get:

• \( \frac{\cos x}{\sec x} = \cos x \cdot \cos x = \cos^2 x \)
• \( \frac{\sin x}{\csc x} = \sin x \cdot \sin x = \sin^2 x \)

Once rewritten, these expressions are much easier to handle, making the verification of identities straightforward. This simplification process is crucial because it turns complex mathematical problems into simpler ones, allowing for deeper insights and solutions.

verifying identities

Verifying trigonometric identities is an essential skill. It involves demonstrating that two sides of an equation are equivalent using known identities and simplifications. The process can seem intimidating, but breaking it down into smaller steps makes it manageable.
Begin by checking both sides of the identity to see if they can be simplified. In our exercise \( \frac{\cos x}{\sec x}+\frac{\sin x}{\csc x}=1 \), simplifying both fractions leads to \( \cos^2 x + \sin^2 x \).

• Next, apply known identities, like the Pythagorean Identity \( \cos^2 x + \sin^2 x = 1 \).
• Finally, confirm if the simplified form equals the other side of the equation.

This process not only reinforces understanding of trigonometric relationships but also strengthens problem-solving skills. Practice is key; the more identities you verify, the quicker and more intuitive the process becomes.