Simplify The Expression: $\[ \frac{1-\cos 2x - \sin X}{\sin 2x - \cos X} = \tan X \\]

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Introduction

In mathematics, trigonometric expressions are a crucial part of various mathematical operations. Simplifying these expressions is essential to understand and solve complex mathematical problems. In this article, we will focus on simplifying the given trigonometric expression: $\frac{1-\cos 2x - \sin x}{\sin 2x - \cos x} = \tan x$. We will use various trigonometric identities and formulas to simplify the expression and arrive at the final result.

Understanding the Given Expression

The given expression is a trigonometric equation that involves the sine and cosine functions. The expression is $\frac{1-\cos 2x - \sin x}{\sin 2x - \cos x} = \tan x$. To simplify this expression, we need to use various trigonometric identities and formulas.

Using Trigonometric Identities

To simplify the given expression, we can use the following trigonometric identities:

• $\cos 2x = 1 - 2\sin^2 x$
• $\sin 2x = 2\sin x \cos x$
• $\tan x = \frac{\sin x}{\cos x}$

We can use these identities to simplify the given expression.

Simplifying the Expression

Using the trigonometric identities mentioned above, we can simplify the given expression as follows:

$\frac{1-\cos 2x - \sin x}{\sin 2x - \cos x} = \frac{1-(1-2\sin^2 x) - \sin x}{2\sin x \cos x - \cos x}$

Simplifying the expression further, we get:

$\frac{1-\cos 2x - \sin x}{\sin 2x - \cos x} = \frac{2\sin^2 x - \sin x}{\sin x (2\cos x - 1)}$

Canceling Common Factors

We can cancel out the common factor of $\sin x$ from the numerator and denominator:

$\frac{2\sin^2 x - \sin x}{\sin x (2\cos x - 1)} = \frac{2\sin x - 1}{2\cos x - 1}$

Using the Double Angle Formula

We can use the double angle formula for sine to simplify the expression further:

$\frac{2\sin x - 1}{2\cos x - 1} = \frac{2\sin x - 1}{2(1 - \sin^2 x) - 1}$

Simplifying the expression further, we get:

$\frac{2\sin x - 1}{2(1 - \sin^2 x) - 1} = \frac{2\sin x - 1}{2\cos^2 x - 1}$

Using the Pythagorean Identity

We can use the Pythagorean identity to simplify the expression further:

$\frac{2\sin x - 1}{2\cos^2 x - 1} = \frac{2\sin x - 1}{\frac{1}{\cos^2 x} - 1}$

Simplifying the expression further, we get:

$\frac{2\sin x - 1}{\frac{1}{\cos^2 x} - 1} = \frac{2\sin x - 1}{\frac{1 - \cos^2 x}{\cos^2 x}}$

Canceling Common Factors

We can cancel out the common factor of $\cos^2 x$ from the denominator:

$\frac{2\sin x - 1}{\frac{1 - \cos^2 x}{\cos^2 x}} = \frac{2\sin x - 1}{\frac{\sin^2 x}{\cos^2 x}}$

Simplifying the expression further, we get:

$\frac{2\sin x - 1}{\frac{\sin^2 x}{\cos^2 x}} = \frac{(2\sin x - 1)\cos^2 x}{\sin^2 x}$

Canceling Common Factors

We can cancel out the common factor of $\sin^2 x$ from the numerator and denominator:

$\frac{(2\sin x - 1)\cos^2 x}{\sin^2 x} = \frac{(2\sin x - 1)\cos x}{\sin x}$

Simplifying the Expression

We can simplify the expression further by canceling out the common factor of $\sin x$ from the numerator and denominator:

$\frac{(2\sin x - 1)\cos x}{\sin x} = 2\cos x - \frac{\cos x}{\sin x}$

Using the Definition of Tangent

We can use the definition of tangent to simplify the expression further:

$2\cos x - \frac{\cos x}{\sin x} = 2\cos x - \tan x$

Simplifying the Expression

We can simplify the expression further by combining like terms:

$2\cos x - \tan x = \frac{2\cos x \sin x - \sin x}{\sin x}$

Simplifying the expression further, we get:

$\frac{2\cos x \sin x - \sin x}{\sin x} = \frac{\sin x (2\cos x - 1)}{\sin x}$

Canceling Common Factors

We can cancel out the common factor of $\sin x$ from the numerator and denominator:

$\frac{\sin x (2\cos x - 1)}{\sin x} = 2\cos x - 1$

Using the Definition of Tangent

We can use the definition of tangent to simplify the expression further:

$2\cos x - 1 = \frac{2\cos x \sin x - \sin x}{\sin x}$

Simplifying the expression further, we get:

$\frac{2\cos x \sin x - \sin x}{\sin x} = \frac{\sin x (2\cos x - 1)}{\sin x}$

Canceling Common Factors

We can cancel out the common factor of $\sin x$ from the numerator and denominator:

$\frac{\sin x (2\cos x - 1)}{\sin x} = 2\cos x - 1$

Conclusion

In this article, we simplified the given trigonometric expression: $\frac{1-\cos 2x - \sin x}{\sin 2x - \cos x} = \tan x$. We used various trigonometric identities and formulas to simplify the expression and arrive at the final result. The final result is $2\cos x - 1 = \tan x$. This result can be used to solve various mathematical problems involving trigonometric expressions.

Final Answer

The final answer is $\boxed{2\cos x - 1 = \tan x}$.

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Introduction

In our previous article, we simplified the given trigonometric expression: $\frac{1-\cos 2x - \sin x}{\sin 2x - \cos x} = \tan x$. We used various trigonometric identities and formulas to simplify the expression and arrive at the final result. In this article, we will answer some frequently asked questions related to the simplification of the given expression.

Q1: What is the main concept behind simplifying the given expression?

A1: The main concept behind simplifying the given expression is to use various trigonometric identities and formulas to simplify the expression and arrive at the final result.

Q2: What are some of the trigonometric identities used to simplify the expression?

A2: Some of the trigonometric identities used to simplify the expression include:

• $\cos 2x = 1 - 2\sin^2 x$
• $\sin 2x = 2\sin x \cos x$
• $\tan x = \frac{\sin x}{\cos x}$

Q3: How do we simplify the expression using the double angle formula?

A3: We can simplify the expression using the double angle formula by substituting $\cos 2x = 1 - 2\sin^2 x$ into the expression.

Q4: How do we simplify the expression using the Pythagorean identity?

A4: We can simplify the expression using the Pythagorean identity by substituting $\cos^2 x = 1 - \sin^2 x$ into the expression.

Q5: What is the final result of simplifying the expression?

A5: The final result of simplifying the expression is $2\cos x - 1 = \tan x$.

Q6: How do we use the definition of tangent to simplify the expression?

A6: We can use the definition of tangent to simplify the expression by substituting $\tan x = \frac{\sin x}{\cos x}$ into the expression.

Q7: What are some of the common factors that can be canceled out in the expression?

A7: Some of the common factors that can be canceled out in the expression include $\sin x$ and $\cos x$.

Q8: How do we simplify the expression using the definition of tangent?

A8: We can simplify the expression using the definition of tangent by substituting $\tan x = \frac{\sin x}{\cos x}$ into the expression.

Q9: What is the importance of simplifying trigonometric expressions?

A9: Simplifying trigonometric expressions is important because it helps us to understand and solve complex mathematical problems.

Q10: How do we apply the simplification of the expression to real-world problems?

A10: We can apply the simplification of the expression to real-world problems by using the final result to solve various mathematical problems involving trigonometric expressions.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the given expression: $\frac{1-\cos 2x - \sin x}{\sin 2x - \cos x} = \tan x$. We used various trigonometric identities and formulas to simplify the expression and arrive at the final result. The final result is $2\cos x - 1 = \tan x$. This result can be used to solve various mathematical problems involving trigonometric expressions.

Final Answer

The final answer is $\boxed{2\cos x - 1 = \tan x}$.