https://r2.erweima.ai/mathbot/image/1727054099082vrc54ued.jpg

The equation provided in the image is:

\[
\frac{\tan(x) - \sin(x)}{\sin^3(x)} = \frac{\sec(x)}{1 + \cos(x)}
\]

### Let's simplify this step by step:

1. **Rewrite everything in terms of sine and cosine:**
   - \(\tan(x) = \frac{\sin(x)}{\cos(x)}\)
   - \(\sec(x) = \frac{1}{\cos(x)}\)

   Now the equation becomes:

   \[
   \frac{\frac{\sin(x)}{\cos(x)} - \sin(x)}{\sin^3(x)} = \frac{\frac{1}{\cos(x)}}{1 + \cos(x)}
   \]

2. **Simplify the left-hand side (LHS):**

   Factor out \(\sin(x)\) from the numerator:

   \[
   \frac{\sin(x) \left( \frac{1}{\cos(x)} - 1 \right)}{\sin^3(x)} = \frac{1 - \cos(x)}{\cos(x)} \cdot \frac{1}{\sin^2(x)}
   \]

   Simplifying further:

   \[
   \frac{1 - \cos(x)}{\cos(x) \sin^2(x)}
   \]

3. **Simplify the right-hand side (RHS):**

   The right-hand side is already simple, so we leave it as:

   \[
   \frac{1}{\cos(x)(1 + \cos(x))}
   \]

4. **Equating both sides:**

   Now we have:

   \[
   \frac{1 - \cos(x)}{\cos(x) \sin^2(x)} = \frac{1}{\cos(x)(1 + \cos(x))}
   \]

5. **Canceling out \(\cos(x)\):**

   Since \(\cos(x) \neq 0\), we can cancel \(\cos(x)\) from both sides:

   \[
   \frac{1 - \cos(x)}{\sin^2(x)} = \frac{1}{1 + \cos(x)}
   \]

6. **Cross-multiply:**

   Cross-multiplying yields:

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

   This simplifies further to:

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

7. **Verify the identity:**

   This is a well-known Pythagorean identity:

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

   Hence, both sides of the equation are equal, verifying that the original equation is **true**.

---

Do you want further details on any part of this solution?

Here are five related questions for practice:
1. How would you prove \(\sin^2(x) + \cos^2(x) = 1\)?
2. What is the derivative of \(\tan(x)\)?
3. How can you express \(\sec(x)\) in terms of \(\sin(x)\) and \(\cos(x)\)?
4. How would you solve \(\tan^2(x) + 1 = \sec^2(x)\)?
5. What is the integral of \(\sec(x)\)?

**Tip:** Always try converting all trigonometric functions into sine and cosine when simplifying complex trigonometric expressions.

Tan(x) - Sen(x) / Sen^3(x) = Sec(x) / (1 + Cos(x))

This guide explains how to simplify the trigonometric equation (tan(x) - sin(x)) / sin^3(x) = sec(x) / (1 + cos(x)). Learn step-by-step techniques using trigonometric identities and algebraic manipulations, ideal for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation