https://file.aiquickdraw.com/mathbot/file/17310131736895jbf6u3i.jpg

The given equation is:

\[
\frac{\tan^2 x}{\sec x} + \cos x = 2
\]

To simplify and solve this, we can proceed by expressing \(\tan x\) and \(\sec x\) in terms of \(\sin x\) and \(\cos x\):

1. **Rewrite \(\tan^2 x\) and \(\sec x\):**
   - \(\tan^2 x = \frac{\sin^2 x}{\cos^2 x}\)
   - \(\sec x = \frac{1}{\cos x}\)

2. **Substitute these into the equation:**
   \[
   \frac{\tan^2 x}{\sec x} = \frac{\frac{\sin^2 x}{\cos^2 x}}{\frac{1}{\cos x}} = \frac{\sin^2 x}{\cos x}
   \]

   So the equation becomes:
   \[
   \frac{\sin^2 x}{\cos x} + \cos x = 2
   \]

3. **Multiply through by \(\cos x\) to clear the denominator:**
   \[
   \sin^2 x + \cos^2 x = 2 \cos x
   \]

4. **Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\):**
   \[
   1 = 2 \cos x
   \]

5. **Solve for \(\cos x\):**
   \[
   \cos x = \frac{1}{2}
   \]

6. **Determine \(x\) values that satisfy \(\cos x = \frac{1}{2}\):**
   \[
   x = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad x = \frac{5\pi}{3} + 2n\pi, \quad n \in \mathbb{Z}
   \]

So, the solutions to the equation are:
\[
x = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad x = \frac{5\pi}{3} + 2n\pi, \quad n \in \mathbb{Z}
\]

Would you like further clarification on any steps or additional examples?

Here are some follow-up questions you may find helpful:
1. Can we solve similar trigonometric equations by using different identities?
2. How does the Pythagorean identity help in simplifying trigonometric expressions?
3. What are other common trigonometric identities useful for solving equations?
4. How can we verify the solutions obtained for trigonometric equations?
5. What are the general solutions for trigonometric equations involving \(\cos x\) and \(\sin x\)?

**Tip:** When working with trigonometric equations, always consider converting to fundamental identities like \(\sin\), \(\cos\), and \(\tan\) to simplify expressions and make solving easier.

Solve the trigonometric equation \(\frac{\tan^2 x}{\sec x} + \cos x = 2\).

Learn how to solve the trigonometric equation \(\frac{\tan^2 x}{\sec x} + \cos x = 2\) step-by-step. This solution involves key trigonometric concepts such as identities and Pythagorean theorem. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation