https://file.aiquickdraw.com/mathbot/file/1726991460243qst0b3ep.jpg

It appears that the same image has been uploaded again. Since we have already solved the problem, here's a summary of the solution for the given equation:

### Given equation:

\[
\sec \theta - \frac{\tan \theta}{\csc \theta} = \cos \theta
\]

### Key steps:

1. **Recall the trigonometric identities** for secant, tangent, and cosecant:
   \[
   \sec \theta = \frac{1}{\cos \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta}
   \]

2. **Substitute these identities** into the equation:
   \[
   \frac{1}{\cos \theta} - \frac{\frac{\sin \theta}{\cos \theta}}{\frac{1}{\sin \theta}} = \cos \theta
   \]

3. **Simplify the expression**:
   \[
   \frac{1}{\cos \theta} - \frac{\sin^2 \theta}{\cos \theta} = \cos \theta
   \]

4. **Use the Pythagorean identity** \(1 - \sin^2 \theta = \cos^2 \theta\):
   \[
   \frac{\cos^2 \theta}{\cos \theta} = \cos \theta
   \]

Thus, the identity is proven.

---

If you would like further assistance or additional explanations, feel free to ask!

Here are five related questions to explore further:

1. How would you approach this identity if the equation had \( \cot \theta \) instead of \( \tan \theta \)?
2. Can you rewrite other trigonometric functions in terms of sine and cosine and prove a different identity?
3. What role does the Pythagorean identity play in simplifying trigonometric expressions?
4. How do the reciprocal identities help in simplifying complex trigonometric expressions?
5. How can you apply this method to more complex trigonometric identities?

**Tip:** Always look for opportunities to use fundamental identities like \( \sin^2 \theta + \cos^2 \theta = 1 \) when working through trigonometric proofs.

Show that sec θ − (tan θ / cosec θ) = cos θ.

Learn how to prove the trigonometric identity sec θ − (tan θ / cosec θ) = cos θ using fundamental identities. This step-by-step solution covers reciprocal and Pythagorean identities, making it 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