Simplify The Expression:$\[ \frac{2 \tan \left(x+180^{\circ}\right) \cos \left(720^{\circ}-x\right)}{\cos \left(450^{\circ}+x\right)}+1 \\]

Introduction

Trigonometric identities are a fundamental concept in mathematics, and they play a crucial role in simplifying complex expressions. In this article, we will focus on simplifying the given expression, which involves various trigonometric functions, including tangent and cosine. We will break down the expression step by step, using trigonometric identities to simplify it.

Understanding the Given Expression

The given expression is:

$$\frac{2 \tan \left(x+180^{\circ}\right) \cos \left(720^{\circ}-x\right)}{\cos \left(450^{\circ}+x\right)}+1$$

This expression involves the tangent and cosine functions, and it includes various angles, such as $x$, $180^{\circ}$, $720^{\circ}$, and $450^{\circ}$. Our goal is to simplify this expression using trigonometric identities.

Simplifying the Expression

To simplify the expression, we will start by using the tangent identity:

$$\tan \left(x+180^{\circ}\right) = \frac{\sin \left(x+180^{\circ}\right)}{\cos \left(x+180^{\circ}\right)}$$

Substituting this identity into the expression, we get:

$$\frac{2 \frac{\sin \left(x+180^{\circ}\right)}{\cos \left(x+180^{\circ}\right)} \cos \left(720^{\circ}-x\right)}{\cos \left(450^{\circ}+x\right)}+1$$

Next, we will use the cosine identity:

$$\cos \left(720^{\circ}-x\right) = \cos \left(720^{\circ}\right) \cos \left(x\right) + \sin \left(720^{\circ}\right) \sin \left(x\right)$$

Substituting this identity into the expression, we get:

$$\frac{2 \frac{\sin \left(x+180^{\circ}\right)}{\cos \left(x+180^{\circ}\right)} \left(\cos \left(720^{\circ}\right) \cos \left(x\right) + \sin \left(720^{\circ}\right) \sin \left(x\right)\right)}{\cos \left(450^{\circ}+x\right)}+1$$

Using Trigonometric Identities to Simplify the Expression

Now, we will use various trigonometric identities to simplify the expression. We will start by using the identity:

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

Substituting this identity into the expression, we get:

$$\frac{2 \frac{-\sin \left(x\right)}{\cos \left(x+180^{\circ}\right)} \left(\cos \left(720^{\circ}\right) \cos \left(x\right) + \sin \left(720^{\circ}\right) \sin \left(x\right)\right)}{\cos \left(450^{\circ}+x\right)}+1$$

Next, we will use the identity:

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

Substituting this identity into the expression, we get:

$$\frac{2 \frac{\sin \left(x\right)}{\cos \left(x\right)} \left(\cos \left(720^{\circ}\right) \cos \left(x\right) + \sin \left(720^{\circ}\right) \sin \left(x\right)\right)}{\cos \left(450^{\circ}+x\right)}+1$$

Simplifying the Expression Further

Now, we will simplify the expression further by using the identity:

$$\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$$

Substituting this identity into the expression, we get:

$$\frac{2 \tan \left(x\right) \left(\cos \left(720^{\circ}\right) \cos \left(x\right) + \sin \left(720^{\circ}\right) \sin \left(x\right)\right)}{\cos \left(450^{\circ}+x\right)}+1$$

Using the Angle Addition Formula for Cosine

Now, we will use the angle addition formula for cosine:

$$\cos \left(a+b\right) = \cos \left(a\right) \cos \left(b\right) - \sin \left(a\right) \sin \left(b\right)$$

Substituting this formula into the expression, we get:

$$\frac{2 \tan \left(x\right) \left(\cos \left(720^{\circ}\right) \cos \left(x\right) + \sin \left(720^{\circ}\right) \sin \left(x\right)\right)}{\cos \left(450^{\circ}\right) \cos \left(x\right) - \sin \left(450^{\circ}\right) \sin \left(x\right)}+1$$

Simplifying the Expression Using Trigonometric Identities

Now, we will simplify the expression using trigonometric identities. We will start by using the identity:

$$\cos \left(720^{\circ}\right) = \cos \left(360^{\circ}\right) = 1$$

Substituting this identity into the expression, we get:

$$\frac{2 \tan \left(x\right) \left(1 \cos \left(x\right) + \sin \left(720^{\circ}\right) \sin \left(x\right)\right)}{\cos \left(450^{\circ}\right) \cos \left(x\right) - \sin \left(450^{\circ}\right) \sin \left(x\right)}+1$$

Using the Identity for $\sin \left(720^{\circ}\right)$

Now, we will use the identity:

$$\sin \left(720^{\circ}\right) = \sin \left(360^{\circ}\right) = 0$$

Substituting this identity into the expression, we get:

$$\frac{2 \tan \left(x\right) \left(\cos \left(x\right)\right)}{\cos \left(450^{\circ}\right) \cos \left(x\right) - \sin \left(450^{\circ}\right) \sin \left(x\right)}+1$$

Simplifying the Expression Further

Now, we will simplify the expression further by using the identity:

$$\cos \left(450^{\circ}\right) = \cos \left(270^{\circ}\right) = 0$$

Substituting this identity into the expression, we get:

$$\frac{2 \tan \left(x\right) \left(\cos \left(x\right)\right)}{-\sin \left(450^{\circ}\right) \sin \left(x\right)}+1$$

Using the Identity for $\sin \left(450^{\circ}\right)$

Now, we will use the identity:

$$\sin \left(450^{\circ}\right) = \sin \left(270^{\circ}\right) = -1$$

Substituting this identity into the expression, we get:

$$\frac{2 \tan \left(x\right) \left(\cos \left(x\right)\right)}{\sin \left(x\right)}+1$$

Simplifying the Expression Using Trigonometric Identities

Now, we will simplify the expression using trigonometric identities. We will start by using the identity:

$$\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$$

Substituting this identity into the expression, we get:

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

Simplifying the Expression Further

Now, we will simplify the expression further by canceling out the common terms:

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

Using the Identity for $\frac{\sin \left(x\right)}{\cos \left(x\right)}$

Now, we will use the identity:

$$\frac{\sin \left(x\right)}{\cos \left(x\right)} = \tan \left(x\right)$$

Substituting this identity into the expression, we get:

$$2 \tan \left(x\right)+1$$

Conclusion

In this article, we have simplified the given expression using various trigonometric identities. We have broken down the expression step by step, using identities such as the tangent identity, the cosine identity, and the angle addition formula for cosine. We have also used various trigonometric identities to simplify the expression further. The final simplified expression is:

$$2 \tan \left(x\right)+1$$

This expression is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and computer science.

Introduction

In our previous article, we simplified the given expression using various trigonometric identities. We broke down the expression step by step, using identities such as the tangent identity, the cosine identity, and the angle addition formula for cosine. In this article, we will answer some of the most frequently asked questions related to the simplification of the expression.

Q: What is the tangent identity?

A: The tangent identity is:

$$\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$$

This identity is used to simplify expressions involving the tangent function.

Q: What is the cosine identity?

A: The cosine identity is:

$$\cos \left(x\right) = \cos \left(a\right) \cos \left(b\right) - \sin \left(a\right) \sin \left(b\right)$$

This identity is used to simplify expressions involving the cosine function.

Q: What is the angle addition formula for cosine?

A: The angle addition formula for cosine is:

$$\cos \left(a+b\right) = \cos \left(a\right) \cos \left(b\right) - \sin \left(a\right) \sin \left(b\right)$$

This formula is used to simplify expressions involving the cosine function.

Q: How do I simplify the expression $\frac{2 \tan \left(x\right) \left(\cos \left(x\right)\right)}{\cos \left(450^{\circ}\right) \cos \left(x\right) - \sin \left(450^{\circ}\right) \sin \left(x\right)}+1$?

A: To simplify this expression, you can use the following steps:

1. Use the identity $\cos \left(450^{\circ}\right) = \cos \left(270^{\circ}\right) = 0$ to simplify the expression.
2. Use the identity $\sin \left(450^{\circ}\right) = \sin \left(270^{\circ}\right) = -1$ to simplify the expression.
3. Use the identity $\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$ to simplify the expression.
4. Cancel out the common terms to simplify the expression.

Q: What is the final simplified expression?

A: The final simplified expression is:

$$2 \tan \left(x\right)+1$$

Q: What are some of the most common trigonometric identities?

A: Some of the most common trigonometric identities include:

• The tangent identity: $\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$
• The cosine identity: $\cos \left(x\right) = \cos \left(a\right) \cos \left(b\right) - \sin \left(a\right) \sin \left(b\right)$
• The angle addition formula for cosine: $\cos \left(a+b\right) = \cos \left(a\right) \cos \left(b\right) - \sin \left(a\right) \sin \left(b\right)$

Q: How do I use trigonometric identities to simplify expressions?

A: To use trigonometric identities to simplify expressions, you can follow these steps:

1. Identify the trigonometric functions involved in the expression.
2. Use the corresponding trigonometric identity to simplify the expression.
3. Cancel out the common terms to simplify the expression.

Q: What are some of the most common applications of trigonometric identities?

A: Some of the most common applications of trigonometric identities include:

• Simplifying expressions involving trigonometric functions
• Solving trigonometric equations
• Finding the values of trigonometric functions for specific angles
• Simplifying expressions involving complex numbers

Conclusion

In this article, we have answered some of the most frequently asked questions related to the simplification of the expression. We have also provided some of the most common trigonometric identities and their applications. By understanding these identities and their applications, you can simplify complex expressions and solve trigonometric equations with ease.