### Proving Trigonometric IdentitiesTo prove the given identity:\[\frac{\sin (x + y)}{\cos x \cos y} = \tan x + \tan y\]Follow these steps:1. **Understand the Formula:** - The left-hand side (LHS) of the identity is $\frac{\sin (x + y)}{\cos x \cos y}$. - The right-hand side (RHS) of the identity is $\tan x + \tan y$.2. **Using the Angle Sum Identity:** - The sine of the sum of two angles can be expressed by the formula: \[ \sin (x + y) = \sin x \cos y + \cos x \sin y \]3. **Substitute in the Identity:** - Replace $\sin (x + y)$ in the original identity with $\sin x \cos y + \cos x \sin y$: \[ \frac{\sin x \cos y + \cos x \sin y}{\cos x \cos y} \]4. **Simplify the Expression:** - Split the fraction into two parts: \[ = \frac{\sin x \cos y}{\cos x \cos y} + \frac{\cos x \sin y}{\cos x \cos y} \] - Simplify each part of the expression: \[ = \frac{\sin x}{\cos x} + \frac{\sin y}{\cos y} \] - Use the definitions of the tangent function: \[ = \tan x + \tan y \]This confirms that:\[\frac{\sin (x + y)}{\cos x \cos y} = \tan x + \tan y\]### Note:- Each Statement must be based on a Rule chosen from the Rule menu. For a detailed understanding of each rule, select the "More Information" Button to the right of the Rule.#### Diagram Explanation:- The lower portion of the image includes a user interface element where users can select the rule used to validate each step of the proof. The "Validate" button allows submission to check if the chosen rule is correct. There is a section titled "Statement" which currently contains the given identity waiting

Name: ### Proving Trigonometric IdentitiesTo prove the given identity:\[\fr
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Description: ### Proving Trigonometric IdentitiesTo prove the given identity:\[\frac{\sin (x + y)}{\cos x \cos y} = \tan x + \tan y\]Follow these steps:1. **Understand the Formula:** - The left-hand side (LHS) of the identity is \(\frac{\sin (x + y)}{\cos x \c

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Answered: Prove the identity. sin (x+y) = tanx+…

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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### Proving Trigonometric Identities

To prove the given identity:

\[
\frac{\sin (x + y)}{\cos x \cos y} = \tan x + \tan y
\]

Follow these steps:

1. **Understand the Formula:**
- The left-hand side (LHS) of the identity is $\frac{\sin (x + y)}{\cos x \cos y}$.
- The right-hand side (RHS) of the identity is $\tan x + \tan y$.

2. **Using the Angle Sum Identity:**
- The sine of the sum of two angles can be expressed by the formula:
\[
\sin (x + y) = \sin x \cos y + \cos x \sin y
\]

3. **Substitute in the Identity:**
- Replace $\sin (x + y)$ in the original identity with $\sin x \cos y + \cos x \sin y$:
\[
\frac{\sin x \cos y + \cos x \sin y}{\cos x \cos y}
\]

4. **Simplify the Expression:**
- Split the fraction into two parts:
\[
= \frac{\sin x \cos y}{\cos x \cos y} + \frac{\cos x \sin y}{\cos x \cos y}
\]
- Simplify each part of the expression:
\[
= \frac{\sin x}{\cos x} + \frac{\sin y}{\cos y}
\]
- Use the definitions of the tangent function:
\[
= \tan x + \tan y
\]

This confirms that:

\[
\frac{\sin (x + y)}{\cos x \cos y} = \tan x + \tan y
\]

### Note:
- Each Statement must be based on a Rule chosen from the Rule menu. For a detailed understanding of each rule, select the "More Information" Button to the right of the Rule.

#### Diagram Explanation:
- The lower portion of the image includes a user interface element where users can select the rule used to validate each step of the proof. The "Validate" button allows submission to check if the chosen rule is correct. There is a section titled "Statement" which currently contains the given identity waiting

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