To solve the equation:

\[
\cos x + \cos x \tan^2 x = \sec x
\]

### Step 1: Simplify the Equation Using Trigonometric Identities
Rewrite the equation in terms of basic trigonometric identities:

1. Recall that \(\sec x = \frac{1}{\cos x}\).
2. Also, \(\tan^2 x = \sec^2 x - 1\) from the Pythagorean identity.

### Step 2: Substitute \(\tan^2 x\) with \(\sec^2 x - 1\)
Rewrite \(\tan^2 x\) in terms of \(\sec x\):

\[
\cos x + \cos x (\sec^2 x - 1) = \sec x
\]

### Step 3: Expand the Terms
Distribute \(\cos x\) in the terms inside the parentheses:

\[
\cos x + \cos x \sec^2 x - \cos x = \sec x
\]

Here, notice that \(\cos x\) and \(-\cos x\) cancel each other out:

\[
\cos x \sec^2 x = \sec x
\]

### Step 4: Simplify Further
Since \(\sec^2 x = \frac{1}{\cos^2 x}\), rewrite the equation as:

\[
\cos x \cdot \frac{1}{\cos^2 x} = \sec x
\]

This simplifies to:

\[
\frac{1}{\cos x} = \sec x
\]

### Step 5: Confirm the Identity
Since \(\sec x = \frac{1}{\cos x}\), both sides of the equation are indeed equal, which confirms the identity. This shows that the equation holds true for values of \(x\) within the domain of the trigonometric functions involved.

Would you like additional details, or do you have any questions about these steps?

Here are some additional related questions:

1. How does the Pythagorean identity involving \(\tan^2 x\) and \(\sec^2 x\) help simplify trigonometric equations?
2. What are the common strategies for solving trigonometric identities?
3. How do we determine the domain restrictions in trigonometric identities?
4. Could we solve similar problems by converting everything to sine and cosine?
5. What are other equivalent forms of trigonometric identities involving \(\sec x\)?

**Tip:** When working with trigonometric identities, expressing everything in terms of \(\sin x\) and \(\cos x\) often makes simplifications easier to see.

Learn how to solve the trigonometric equation cos x + cos x tan^2x = secx by using basic trigonometric identities such as sec x = 1/cos x and the Pythagorean identity. This detailed solution explains each step and simplification, suitable for high school students studying trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation