Simplify The Expression:${ 1 - \frac{\sin 2\left(180 {\circ} + X\right)}{\cos\left(x - 180^{\circ}\right) \cdot \cos\left(x - 360^{\circ}\right)} }$

Introduction

Trigonometric expressions can be complex and challenging to simplify, especially when dealing with multiple angles and trigonometric functions. In this article, we will focus on simplifying a specific trigonometric expression involving sine and cosine functions. We will break down the expression into manageable steps, using trigonometric identities and properties to simplify it.

The Given Expression

The given expression is:

${ 1 - \frac{\sin^2\left(180^{\circ} + x\right)}{\cos\left(x - 180^{\circ}\right) \cdot \cos\left(x - 360^{\circ}\right)} }$

This expression involves sine and cosine functions, as well as angle addition and subtraction. Our goal is to simplify this expression using trigonometric identities and properties.

Step 1: Simplify the Sine Function

The first step is to simplify the sine function in the numerator. We can use the angle addition identity for sine:

${ \sin(A + B) = \sin A \cos B + \cos A \sin B }$

In this case, we have:

${ \sin(180^{\circ} + x) = \sin 180^{\circ} \cos x + \cos 180^{\circ} \sin x }$

Since $\sin 180^{\circ} = 0$ and $\cos 180^{\circ} = -1$, we can simplify the expression to:

${ \sin(180^{\circ} + x) = -\sin x }$

Step 2: Simplify the Cosine Functions

The next step is to simplify the cosine functions in the denominator. We can use the angle subtraction identity for cosine:

${ \cos(A - B) = \cos A \cos B + \sin A \sin B }$

In this case, we have:

${ \cos(x - 180^{\circ}) = \cos x \cos 180^{\circ} + \sin x \sin 180^{\circ} }$

Since $\cos 180^{\circ} = -1$ and $\sin 180^{\circ} = 0$, we can simplify the expression to:

${ \cos(x - 180^{\circ}) = -\cos x }$

Similarly, we can simplify the second cosine function:

${ \cos(x - 360^{\circ}) = \cos x \cos 360^{\circ} + \sin x \sin 360^{\circ} }$

Since $\cos 360^{\circ} = 1$ and $\sin 360^{\circ} = 0$, we can simplify the expression to:

${ \cos(x - 360^{\circ}) = \cos x }$

Step 3: Combine the Simplified Expressions

Now that we have simplified the sine and cosine functions, we can combine the expressions to simplify the original expression. We have:

${ 1 - \frac{\sin^2\left(180^{\circ} + x\right)}{\cos\left(x - 180^{\circ}\right) \cdot \cos\left(x - 360^{\circ}\right)} }$

Substituting the simplified expressions, we get:

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

Simplifying further, we get:

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

Using the identity $\sin^2 x + \cos^2 x = 1$, we can rewrite the expression as:

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

Simplifying further, we get:

${ 1 + \frac{1 - \cos^2 x}{\cos^2 x} }$

Using the identity $\sin^2 x + \cos^2 x = 1$, we can rewrite the expression as:

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

Step 4: Final Simplification

The final step is to simplify the expression using the identity $\tan^2 x + 1 = \sec^2 x$. We can rewrite the expression as:

${ 1 + \tan^2 x }$

Using the identity $\tan^2 x + 1 = \sec^2 x$, we can rewrite the expression as:

${ \sec^2 x }$

Therefore, the simplified expression is:

${ \sec^2 x }$

Conclusion

In this article, we simplified a trigonometric expression involving sine and cosine functions. We broke down the expression into manageable steps, using trigonometric identities and properties to simplify it. The final simplified expression is $\sec^2 x$. This expression can be used to solve problems involving trigonometric functions and identities.

Common Trigonometric Identities

Here are some common trigonometric identities that we used in this article:

• $\sin(A + B) = \sin A \cos B + \cos A \sin B$
• $\cos(A - B) = \cos A \cos B + \sin A \sin B$
• $\sin^2 x + \cos^2 x = 1$
• $\tan^2 x + 1 = \sec^2 x$

These identities can be used to simplify trigonometric expressions and solve problems involving trigonometric functions.

Real-World Applications

Trigonometric expressions and identities have many real-world applications in fields such as physics, engineering, and computer science. Some examples include:

• Physics: Trigonometric expressions are used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Trigonometric expressions are used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
• Computer Science: Trigonometric expressions are used in computer graphics, game development, and other areas of computer science.

Q: What is the purpose of simplifying trigonometric expressions?

A: The purpose of simplifying trigonometric expressions is to make them easier to work with and understand. Simplified expressions can be used to solve problems involving trigonometric functions and identities.

Q: What are some common trigonometric identities that can be used to simplify expressions?

A: Some common trigonometric identities that can be used to simplify expressions include:

• $\sin(A + B) = \sin A \cos B + \cos A \sin B$
• $\cos(A - B) = \cos A \cos B + \sin A \sin B$
• $\sin^2 x + \cos^2 x = 1$
• $\tan^2 x + 1 = \sec^2 x$

Q: How can I simplify a trigonometric expression involving sine and cosine functions?

A: To simplify a trigonometric expression involving sine and cosine functions, you can use the following steps:

1. Simplify the sine function using the angle addition identity.
2. Simplify the cosine functions using the angle subtraction identity.
3. Combine the simplified expressions to simplify the original expression.

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are both trigonometric functions that describe the relationship between the angles and side lengths of triangles. The sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse, while the cosine function is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Q: How can I use trigonometric identities to solve problems involving trigonometric functions?

A: To use trigonometric identities to solve problems involving trigonometric functions, you can follow these steps:

1. Identify the trigonometric function(s) involved in the problem.
2. Use the appropriate trigonometric identity to simplify the expression.
3. Solve the simplified expression to find the solution to the problem.

Q: What are some real-world applications of trigonometric expressions and identities?

A: Some real-world applications of trigonometric expressions and identities include:

• Physics: Trigonometric expressions are used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Trigonometric expressions are used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
• Computer Science: Trigonometric expressions are used in computer graphics, game development, and other areas of computer science.

Q: How can I practice simplifying trigonometric expressions?

A: To practice simplifying trigonometric expressions, you can try the following:

• Start with simple expressions and gradually work your way up to more complex ones.
• Use online resources, such as trigonometric calculators and worksheets, to practice simplifying expressions.
• Work with a partner or tutor to practice simplifying expressions and get feedback on your work.

Q: What are some common mistakes to avoid when simplifying trigonometric expressions?

A: Some common mistakes to avoid when simplifying trigonometric expressions include:

• Forgetting to use the correct trigonometric identity.
• Not simplifying the expression enough.
• Making errors when simplifying the expression.

By following these tips and practicing simplifying trigonometric expressions, you can become more confident and proficient in using trigonometric identities to solve problems involving trigonometric functions.