The given expression is:

\[
x \tan(y) \sin(y) + x \cos(y)
\]

Let's simplify it step by step.

### Step 1: Express \(\tan(y)\) in terms of \(\sin(y)\) and \(\cos(y)\).

We know that:
\[
\tan(y) = \frac{\sin(y)}{\cos(y)}
\]

Substitute this into the original expression:

\[
x \left( \frac{\sin(y)}{\cos(y)} \right) \sin(y) + x \cos(y)
\]

### Step 2: Simplify the first term.

\[
x \cdot \frac{\sin^2(y)}{\cos(y)} + x \cos(y)
\]

### Step 3: Factor out \(x\).

\[
x \left( \frac{\sin^2(y)}{\cos(y)} + \cos(y) \right)
\]

### Step 4: Combine the terms over a common denominator.

The common denominator is \(\cos(y)\), so we rewrite the expression as:

\[
x \left( \frac{\sin^2(y) + \cos^2(y)}{\cos(y)} \right)
\]

### Step 5: Use the Pythagorean identity.

We know that:
\[
\sin^2(y) + \cos^2(y) = 1
\]

So, the expression simplifies to:

\[
x \left( \frac{1}{\cos(y)} \right)
\]

### Final simplified form:

\[
x \sec(y)
\]

The simplified expression is \(x \sec(y)\).

Would you like more details, or do you have any further questions?

Here are 5 related questions you might find interesting:

1. How does the Pythagorean identity help in simplifying trigonometric expressions?
2. What is the relationship between \(\tan(y)\), \(\sin(y)\), and \(\cos(y)\)?
3. Can you provide examples of other expressions simplified using the Pythagorean identity?
4. What is the secant function \(\sec(y)\), and how is it used in trigonometry?
5. How would you differentiate the expression \(x \sec(y)\) with respect to \(x\)?

**Tip:** When simplifying trigonometric expressions, always look for identities like \(\sin^2(x) + \cos^2(x) = 1\) that can reduce the complexity.

Learn how to simplify the trigonometric expression xtan(y)sin(y) + xcos(y) step-by-step. This problem involves key concepts like the Pythagorean identity, secant function, and the relationship between sine, cosine, and tangent. Ideal for high school 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