Simplify Sin ⁡ X Cos ⁡ X + 1 − Cos ⁡ X + 1 Sin ⁡ X \frac{\sin X}{\cos X + 1} - \frac{\cos X + 1}{\sin X} C O S X + 1 S I N X ​ − S I N X C O S X + 1 ​ .

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Introduction

Trigonometric simplification is a crucial aspect of mathematics, and it often involves manipulating expressions involving trigonometric functions. In this article, we will focus on simplifying the given expression $\frac{\sin x}{\cos x + 1} - \frac{\cos x + 1}{\sin x}$ using various trigonometric identities and techniques.

Understanding the Expression

The given expression involves two fractions, each with a trigonometric function in the numerator and a different trigonometric function in the denominator. To simplify this expression, we need to find a common denominator and then combine the fractions.

Finding a Common Denominator

To find a common denominator, we need to multiply the denominators of both fractions. In this case, the denominators are $\cos x + 1$ and $\sin x$. We can multiply both denominators by $\sin x$ and $\cos x + 1$ to get a common denominator of $(\cos x + 1)\sin x$.

Simplifying the Expression

Now that we have a common denominator, we can rewrite the expression as follows:

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

Expanding the Numerator

To simplify the numerator, we need to expand the squared term $(\cos x + 1)^2$. Using the formula $(a + b)^2 = a^2 + 2ab + b^2$, we get:

$$(\cos x + 1)^2 = \cos^2 x + 2\cos x + 1$$

Simplifying the Numerator

Now that we have expanded the squared term, we can simplify the numerator by combining like terms:

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

Using Trigonometric Identities

We can simplify the expression further by using the trigonometric identity $\sin^2 x - \cos^2 x = -\cos 2x$. Substituting this identity into the numerator, we get:

$$-\cos 2x - 2\cos x - 1$$

Simplifying the Expression

Now that we have simplified the numerator, we can rewrite the expression as follows:

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

Factoring the Numerator

We can factor the numerator by grouping terms:

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

Simplifying the Expression

Now that we have factored the numerator, we can rewrite the expression as follows:

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

Canceling Common Factors

We can cancel the common factor of $-1$ in the numerator and denominator:

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

Using Trigonometric Identities

We can simplify the expression further by using the trigonometric identity $\cos 2x = 2\cos^2 x - 1$. Substituting this identity into the numerator, we get:

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

Simplifying the Expression

Now that we have simplified the numerator, we can rewrite the expression as follows:

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

Factoring the Numerator

We can factor the numerator by grouping terms:

$$2\cos^2 x + 2\cos x = 2\cos x(\cos x + 1)$$

Simplifying the Expression

Now that we have factored the numerator, we can rewrite the expression as follows:

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

Canceling Common Factors

We can cancel the common factor of $\cos x + 1$ in the numerator and denominator:

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

Using Trigonometric Identities

We can simplify the expression further by using the trigonometric identity $\frac{\cos x}{\sin x} = \cot x$. Substituting this identity into the expression, we get:

$$\frac{2\cos x}{\sin x} = 2\cot x$$

Conclusion

In this article, we simplified the given expression $\frac{\sin x}{\cos x + 1} - \frac{\cos x + 1}{\sin x}$ using various trigonometric identities and techniques. We started by finding a common denominator and then combined the fractions. We then expanded the numerator, simplified the expression, and used trigonometric identities to simplify the expression further. Finally, we canceled common factors and used trigonometric identities to simplify the expression to its final form, $2\cot x$.

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Introduction

In our previous article, we simplified the given expression $\frac{\sin x}{\cos x + 1} - \frac{\cos x + 1}{\sin x}$ using various trigonometric identities and techniques. In this article, we will answer some common questions that readers may have about the simplification process.

Q: What is the common denominator of the two fractions?

A: The common denominator of the two fractions is $(\cos x + 1)\sin x$.

Q: How do you expand the numerator of the expression?

A: To expand the numerator, we need to multiply the squared term $(\cos x + 1)^2$ using the formula $(a + b)^2 = a^2 + 2ab + b^2$. This gives us $(\cos x + 1)^2 = \cos^2 x + 2\cos x + 1$.

Q: What is the simplified form of the numerator?

A: The simplified form of the numerator is $\sin^2 x - \cos^2 x - 2\cos x - 1$.

Q: How do you simplify the expression further?

A: We can simplify the expression further by using the trigonometric identity $\sin^2 x - \cos^2 x = -\cos 2x$. Substituting this identity into the numerator, we get $-\cos 2x - 2\cos x - 1$.

Q: What is the final simplified form of the expression?

A: The final simplified form of the expression is $2\cot x$.

Q: What trigonometric identities were used in the simplification process?

A: The following trigonometric identities were used in the simplification process:

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

Q: What is the importance of finding a common denominator in the simplification process?

A: Finding a common denominator is crucial in the simplification process as it allows us to combine the fractions and simplify the expression.

Q: What is the significance of canceling common factors in the simplification process?

A: Canceling common factors is important in the simplification process as it helps to simplify the expression and make it easier to understand.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is $2\cot x$.

Conclusion

In this article, we answered some common questions that readers may have about the simplification process of the given expression $\frac{\sin x}{\cos x + 1} - \frac{\cos x + 1}{\sin x}$. We provided detailed explanations and examples to help readers understand the simplification process and the importance of finding a common denominator and canceling common factors.

Additional Resources

For more information on trigonometric identities and simplification techniques, please refer to the following resources:

• Trigonometric Identities: A Comprehensive Guide
• Simplifying Trigonometric Expressions: A Step-by-Step Guide
• Trigonometry: A Beginner's Guide

Final Thoughts

Simplifying trigonometric expressions can be a challenging task, but with the right techniques and tools, it can be made easier. By understanding the importance of finding a common denominator and canceling common factors, readers can simplify complex expressions and make them easier to understand.