https://file.aiquickdraw.com/mathbot/file/1726016288532z5iqomka.jpg

Let's go through the steps of proving the identity \( \cot(x) + \tan(x) = \csc(x) \sec(x) \) by following the blanks in the image:

### Step 1:
The identity starts with:
\[
\cot(x) + \tan(x) = \frac{\cos(x)}{\sin(x)} + \frac{\sin(x)}{\cos(x)}
\]
So, the first blank will be:
\[
\frac{\cos(x)}{\sin(x)} + \frac{\sin(x)}{\cos(x)}
\]

### Step 2:
Now, to combine the two fractions:
\[
\frac{\cos(x)}{\sin(x)} + \frac{\sin(x)}{\cos(x)} = \frac{\cos^2(x) + \sin^2(x)}{\sin(x)\cos(x)}
\]
Since \( \cos^2(x) + \sin^2(x) = 1 \), the second blank will be:
\[
\frac{1}{\sin(x) \cos(x)}
\]

### Step 3:
The expression simplifies to:
\[
\frac{1}{\sin(x) \cos(x)}
\]

### Step 4:
This can be factored as:
\[
\left( \frac{1}{\sin(x)} \right) \left( \frac{1}{\cos(x)} \right)
\]
Thus, the next blank will be:
\[
\left( \csc(x) \right) \left( \sec(x) \right)
\]

### Conclusion:
Now, the final form is:
\[
\csc(x) \sec(x)
\]

This completes the proof that \( \cot(x) + \tan(x) = \csc(x) \sec(x) \).

Would you like any further explanation on any of the steps, or do you have any questions?

### 5 Related Questions:
1. How can you simplify expressions involving \( \cot(x) \) and \( \tan(x) \)?
2. What is the Pythagorean identity, and how is it used in trigonometric proofs?
3. How can you combine two fractions with different denominators?
4. What are the definitions of \( \csc(x) \) and \( \sec(x) \) in terms of sine and cosine?
5. Why is \( \cos^2(x) + \sin^2(x) = 1 \) an important identity in trigonometry?

### Tip:
When solving trigonometric identities, it helps to express all terms using sine and cosine to simplify the process of combining fractions.

Prove the trigonometric identity cot(x) + tan(x) = csc(x) sec(x).

Learn how to prove the trigonometric identity cot(x) + tan(x) = csc(x) sec(x) using simplification techniques and core trigonometric formulas such as cotangent, tangent, cosecant, and secant. This solution covers essential knowledge for students in Precalculus (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