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

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Introduction

In mathematics, trigonometric identities are essential for solving various problems in trigonometry, calculus, and other branches of mathematics. One of the fundamental trigonometric identities is the double-angle formula for cosine, which states that $\cos 2x = 1 - 2\sin^2 x$ or $\cos 2x = 2\cos^2 x - 1$. In this article, we will simplify the given expression $\frac{1-\cos 2x-\sin x}{\sin 2x-\cos x}=\tan x$ using trigonometric identities and provide a step-by-step solution.

Trigonometric Identities

Before we proceed with simplifying the given expression, let's recall some essential trigonometric identities:

• Double-angle formula for cosine: $\cos 2x = 1 - 2\sin^2 x$ or $\cos 2x = 2\cos^2 x - 1$
• Double-angle formula for sine: $\sin 2x = 2\sin x \cos x$
• Pythagorean identity: $\sin^2 x + \cos^2 x = 1$

Simplifying the Expression

To simplify the given expression, we will use the double-angle formula for cosine and the Pythagorean identity.

Step 1: Simplify the numerator

The numerator of the given expression is $1 - \cos 2x - \sin x$. We can simplify this expression using the double-angle formula for cosine:

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

Simplifying further, we get:

$$1 - \cos 2x - \sin x = 2 - 2\cos^2 x - \sin x$$

Step 2: Simplify the denominator

The denominator of the given expression is $\sin 2x - \cos x$. We can simplify this expression using the double-angle formula for sine:

$$\sin 2x - \cos x = 2\sin x \cos x - \cos x$$

Simplifying further, we get:

$$\sin 2x - \cos x = \cos x(2\sin x - 1)$$

Step 3: Simplify the expression

Now that we have simplified the numerator and denominator, we can simplify the expression:

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

Canceling Common Factors

We can cancel the common factor of $2\sin x - 1$ in the numerator and denominator:

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

Using the Pythagorean identity, we can simplify the expression further:

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

Factoring Out Common Factors

We can factor out the common factor of $\sin x$ in the numerator:

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

Canceling Common Factors

We can cancel the common factor of $2\sin x - 1$ in the numerator and denominator:

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

Simplifying the Expression

The expression $\frac{\sin x}{\cos x}$ is equal to $\tan x$. Therefore, we have:

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

Conclusion

In this article, we simplified the given expression $\frac{1-\cos 2x-\sin x}{\sin 2x-\cos x}=\tan x$ using trigonometric identities. We used the double-angle formula for cosine and the Pythagorean identity to simplify the expression. The final simplified expression is $\tan x$, which confirms the given expression.

References

• [1] "Trigonometry" by Michael Corral, 2015.
• [2] "Calculus" by Michael Spivak, 2008.
• [3] "Trigonometric Identities" by Paul Dawkins, 2013.

Further Reading

• [1] "Trigonometric Identities" by Paul Dawkins, 2013.
• [2] "Calculus" by Michael Spivak, 2008.
• [3] "Trigonometry" by Michael Corral, 2015.

Keywords

• Trigonometric identities
• Double-angle formula for cosine
• Pythagorean identity
• Simplifying expressions
• Trigonometry
• Calculus
  

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Introduction

In our previous article, we simplified the given expression $\frac{1-\cos 2x-\sin x}{\sin 2x-\cos x}=\tan x$ using trigonometric identities. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q1: What is the double-angle formula for cosine?

A1: The double-angle formula for cosine is $\cos 2x = 1 - 2\sin^2 x$ or $\cos 2x = 2\cos^2 x - 1$.

Q2: How do you simplify the numerator of the given expression?

A2: To simplify the numerator, we can use the double-angle formula for cosine. We can rewrite the numerator as $1 - \cos 2x - \sin x = 1 - (2\cos^2 x - 1) - \sin x$. Simplifying further, we get $1 - \cos 2x - \sin x = 2 - 2\cos^2 x - \sin x$.

Q3: How do you simplify the denominator of the given expression?

A3: To simplify the denominator, we can use the double-angle formula for sine. We can rewrite the denominator as $\sin 2x - \cos x = 2\sin x \cos x - \cos x$. Simplifying further, we get $\sin 2x - \cos x = \cos x(2\sin x - 1)$.

Q4: How do you simplify the expression?

A4: To simplify the expression, we can cancel the common factor of $2\sin x - 1$ in the numerator and denominator. We can rewrite the expression as $\frac{2 - 2\cos^2 x - \sin x}{\cos x(2\sin x - 1)} = \frac{2(1 - \cos^2 x) - \sin x}{\cos x(2\sin x - 1)}$. Using the Pythagorean identity, we can simplify the expression further as $\frac{2\sin^2 x - \sin x}{\cos x(2\sin x - 1)}$.

Q5: How do you factor out common factors in the expression?

A5: We can factor out the common factor of $\sin x$ in the numerator. We can rewrite the expression as $\frac{2\sin^2 x - \sin x}{\cos x(2\sin x - 1)} = \frac{\sin x(2\sin x - 1)}{\cos x(2\sin x - 1)}$.

Q6: How do you cancel common factors in the expression?

A6: We can cancel the common factor of $2\sin x - 1$ in the numerator and denominator. We can rewrite the expression as $\frac{\sin x(2\sin x - 1)}{\cos x(2\sin x - 1)} = \frac{\sin x}{\cos x}$.

Q7: What is the final simplified expression?

A7: The final simplified expression is $\frac{\sin x}{\cos x}$, which is equal to $\tan x$.

Q8: What trigonometric identities were used to simplify the expression?

A8: We used the double-angle formula for cosine and the Pythagorean identity to simplify the expression.

Q9: What is the significance of the simplified expression?

A9: The simplified expression $\tan x$ is an important trigonometric identity that can be used to solve various problems in trigonometry and calculus.

Q10: Where can I find more information on trigonometric identities?

A10: You can find more information on trigonometric identities in textbooks such as "Trigonometry" by Michael Corral, 2015, and "Calculus" by Michael Spivak, 2008.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $\frac{1-\cos 2x-\sin x}{\sin 2x-\cos x}=\tan x$. We used trigonometric identities such as the double-angle formula for cosine and the Pythagorean identity to simplify the expression. The final simplified expression is $\tan x$, which is an important trigonometric identity that can be used to solve various problems in trigonometry and calculus.

References

• [1] "Trigonometry" by Michael Corral, 2015.
• [2] "Calculus" by Michael Spivak, 2008.
• [3] "Trigonometric Identities" by Paul Dawkins, 2013.

Further Reading

• [1] "Trigonometric Identities" by Paul Dawkins, 2013.
• [2] "Calculus" by Michael Spivak, 2008.
• [3] "Trigonometry" by Michael Corral, 2015.

Keywords

• Trigonometric identities
• Double-angle formula for cosine
• Pythagorean identity
• Simplifying expressions
• Trigonometry
• Calculus