(1)/(2)(\cot x+\tan )=\csc 2x                   this is the guide for what to do in 1st equation isend Summary of Rules Used:
Pythagorean Identity: 
1
+
tan
⁡
2
𝑥
=
sec
⁡
2
𝑥
1+tan 
2
 x=sec 
2
 x
Reciprocal Identity: 
sec
⁡
𝑥
=
1
cos
⁡
𝑥
secx= 
cosx
1
​
  and 
csc
⁡
𝑥
=
1
sin
⁡
𝑥
cscx= 
sinx
1
​
 
Quotient Identity: 
tan
⁡
𝑥
=
sin
⁡
𝑥
cos
⁡
𝑥
tanx= 
cosx
sinx
​
 
Double-Angle Identity: 
sin
⁡
(
2
𝑥
)
=
2
sin
⁡
𝑥
cos
⁡
𝑥
sin(2x)=2sinxcosx

Let's break down the given equation and simplify it using the provided trigonometric identities.

### Given Equation:

\[
\frac{1}{2}(\cot x + \tan x) = \csc(2x)
\]

### Step-by-Step Solution:

#### 1. Express \(\cot x\) and \(\tan x\) in terms of \(\sin x\) and \(\cos x\):
Using the **quotient identity**:
- \(\tan x = \frac{\sin x}{\cos x}\)
- \(\cot x = \frac{\cos x}{\sin x}\)

Substitute these into the equation:

\[
\frac{1}{2} \left( \frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} \right) = \csc(2x)
\]

#### 2. Simplify the left-hand side (LHS):
The LHS becomes:

\[
\frac{1}{2} \left( \frac{\cos^2 x + \sin^2 x}{\sin x \cos x} \right)
\]

Using the **Pythagorean identity** \(\sin^2 x + \cos^2 x = 1\), this simplifies to:

\[
\frac{1}{2} \left( \frac{1}{\sin x \cos x} \right)
\]

\[
\frac{1}{2} \cdot \frac{1}{\sin x \cos x} = \frac{1}{2 \sin x \cos x}
\]

#### 3. Apply the **double-angle identity** for sine:
\[
\sin(2x) = 2 \sin x \cos x
\]

Thus, the LHS becomes:

\[
\frac{1}{\sin(2x)} = \csc(2x)
\]

#### 4. Conclusion:
The left-hand side equals the right-hand side of the equation:

\[
\frac{1}{\sin(2x)} = \csc(2x)
\]

This confirms that the equation is true.

---

### Summary of Steps:
1. Expressed \(\cot x\) and \(\tan x\) in terms of sine and cosine.
2. Simplified the expression using the Pythagorean identity.
3. Applied the double-angle identity for sine.
4. Confirmed that both sides of the equation are equal.

Would you like a more detailed explanation on any of these steps?

### Related Questions:
1. How can you derive the Pythagorean identity from the unit circle?
2. How is the double-angle identity for sine related to the product of \(\sin x\) and \(\cos x\)?
3. Can you solve for \(x\) if given a specific value for \(\csc(2x)\)?
4. How would this equation change if \(\cot x\) and \(\tan x\) were expressed in terms of angles in different quadrants?
5. What is the graphical interpretation of the equation \(\frac{1}{2}(\cot x + \tan x) = \csc(2x)\)?

### Tip:
When simplifying trigonometric equations, it's often helpful to convert everything into sine and cosine terms to apply identities easily.

Learn how to simplify the trigonometric equation (1/2)(cot x + tan x) = csc(2x) using key trigonometric identities, such as the Pythagorean identity, reciprocal identity, quotient identity, and double-angle identity. Suitable for high school students in Grades 10-12.

Math Problem Statement

(1)/(2)(\cot x+\tan )=\csc 2x this is the guide for what to do in 1st equation isend Summary of Rules Used: Pythagorean Identity: 1 + tan ⁡ 2 𝑥

sec ⁡ 2 𝑥 1+tan 2 x=sec 2 x Reciprocal Identity: sec ⁡ 𝑥

1 cos ⁡ 𝑥 secx= cosx 1 and csc ⁡ 𝑥

Quotient Identity: tan ⁡ 𝑥

Double-Angle Identity: sin ⁡ ( 2 𝑥 )

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation