**Title**: Proving the Trigonometric Identity**Objective**: Learn and understand how to prove the given trigonometric identity.**Given Identity**:\[ \frac{\tan x + \cot x}{\csc x \cos x} = \sec^2 x \]**Task**: Drag an expression to each box to correctly complete the proof.### Steps to Complete the Proof:- **Step 1**: \[ \frac{\tan x + \cot x}{\csc x \cos x} = \sec^2 x \] To prove the identity, first break down the given components. - **Step 2**: Simplify $\tan x$ and $\cot x$: \[ \tan x = \frac{\sin x}{\cos x} \quad \text{and} \quad \cot x = \frac{\cos x}{\sin x} \]- **Step 3**: Combine the fractions under a common denominator: \[ \tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} \]- **Step 4**: Use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$: \[ \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x} \]- **Step 5**: Simplify $\csc x \cos x$: \[ \csc x = \frac{1}{\sin x} \] So, \[ \frac{\tan x + \cot x}{\csc x \cos x} = \frac{\frac{1}{\sin x \cos x}}{\frac{1}{\sin x} \cos x} = \frac{1}{\cos^2 x} \]- **Step 6**: Recognize that $\frac{1}{\cos^2 x} = \sec^2 x$: \[ \frac{1}{\cos^2 x} = \sec^2 x \] This image appears to show a series of mathematical expressions involving trigonometric functions. Below is the transcription and explanation of the expressions presented, step-by-step:1. Initial Expression:\[ \frac{1}{\sin x \cos x} \cdot \frac{\sin x}{\cos x} \]2. Combination of Fractions:\[ \frac{1}{\sin x \cos x} \cdot \frac{\sin x}{\cos x} = \frac{1 \cdot \sin x}{\sin x \cos x \cdot \cos x} = \frac{\sin x}{\sin x \cos^2 x} \]3. Simplification:\[ \frac{\sin x}{\sin x \cos^2 x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos^2 x} \]4. Trigonometric Identity:\[ \text{Since} \ \sin^2 x + \cos^2 x = 1, \ \text{the expression becomes} \ \frac{1}{\sin x \cos^2 x} \]5. Further Simplification:\[ \frac{1}{\sin x \cos^2 x} \cdot \cos x = \frac{\cos x}{\sin x \cos^2 x} = \frac{\cos x}{\sin x \cos x} \cdot \frac{1}{\cos x} \]6. Simplifying the Multiplication:\[ \frac{\cos x}{\sin x \cos x} \cdot \frac{1}{\cos x} = \frac{1}{\sin x \cos x} \]7. Final Simplified Expression:\[ \frac{1}{\sin x \cos x} \]Throughout the steps, the process involves breaking down and simplifying the trigonometric expressions, leveraging trigonometric identities, and combining fractions appropriately. The expression is continually simplified until it reaches the final result: $\frac{1}{\sin x \cos x}$.

Answered: Prove the trigonometric identity. tan…

Prove the trigonometric identity. tan x+cot sec? Csc x cos x Drag an expression to each box to correctly complete the proof. tan x+cot x sec2 x Csc x cos x - sec²x sec2x sec²x - sec?x sec²x

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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Question

**Title**: Proving the Trigonometric Identity

**Objective**: Learn and understand how to prove the given trigonometric identity.

**Given Identity**:
\[ \frac{\tan x + \cot x}{\csc x \cos x} = \sec^2 x \]

**Task**: Drag an expression to each box to correctly complete the proof.

### Steps to Complete the Proof:

- **Step 1**:
\[ \frac{\tan x + \cot x}{\csc x \cos x} = \sec^2 x \]
To prove the identity, first break down the given components.

- **Step 2**:
Simplify $\tan x$ and $\cot x$:
\[ \tan x = \frac{\sin x}{\cos x} \quad \text{and} \quad \cot x = \frac{\cos x}{\sin x} \]

- **Step 3**:
Combine the fractions under a common denominator:
\[ \tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} \]

- **Step 4**:
Use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$:
\[ \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x} \]

- **Step 5**:
Simplify $\csc x \cos x$:
\[ \csc x = \frac{1}{\sin x} \]
So,
\[ \frac{\tan x + \cot x}{\csc x \cos x} = \frac{\frac{1}{\sin x \cos x}}{\frac{1}{\sin x} \cos x} = \frac{1}{\cos^2 x} \]

- **Step 6**:
Recognize that $\frac{1}{\cos^2 x} = \sec^2 x$:
\[ \frac{1}{\cos^2 x} = \sec^2 x \]

This image appears to show a series of mathematical expressions involving trigonometric functions. Below is the transcription and explanation of the expressions presented, step-by-step:

1. Initial Expression:
\[ \frac{1}{\sin x \cos x} \cdot \frac{\sin x}{\cos x} \]

2. Combination of Fractions:
\[ \frac{1}{\sin x \cos x} \cdot \frac{\sin x}{\cos x} = \frac{1 \cdot \sin x}{\sin x \cos x \cdot \cos x} = \frac{\sin x}{\sin x \cos^2 x} \]

3. Simplification:
\[ \frac{\sin x}{\sin x \cos^2 x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos^2 x} \]

4. Trigonometric Identity:
\[ \text{Since} \ \sin^2 x + \cos^2 x = 1, \ \text{the expression becomes} \ \frac{1}{\sin x \cos^2 x} \]

5. Further Simplification:
\[ \frac{1}{\sin x \cos^2 x} \cdot \cos x = \frac{\cos x}{\sin x \cos^2 x} = \frac{\cos x}{\sin x \cos x} \cdot \frac{1}{\cos x} \]

6. Simplifying the Multiplication:
\[ \frac{\cos x}{\sin x \cos x} \cdot \frac{1}{\cos x} = \frac{1}{\sin x \cos x} \]

7. Final Simplified Expression:
\[ \frac{1}{\sin x \cos x} \]

Throughout the steps, the process involves breaking down and simplifying the trigonometric expressions, leveraging trigonometric identities, and combining fractions appropriately. The expression is continually simplified until it reaches the final result: $\frac{1}{\sin x \cos x}$.

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