Without Using A Calculator, Simplify The Following Expression Fully:$\[ \frac{\sin \left(180^{\circ}-x\right) \cdot \tan \left(x-180^{\circ}\right) \cdot \cos \left(360^{\circ}+x\right)}{\sin ^2\left(180^{\circ}+x\right)+\sin

**Without using a calculator, simplify the following expression fully**

Understanding the Expression

The given expression involves trigonometric functions such as sine, tangent, and cosine. To simplify this expression, we need to apply various trigonometric identities and formulas.

Step 1: Simplify the Expression using Trigonometric Identities

The given expression is:

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

We can start by simplifying the expression using trigonometric identities.

Simplifying the Sine Terms

We know that $\sin (180^{\circ} - x) = \sin x$ and $\sin (180^{\circ} + x) = -\sin x$. We can substitute these values into the expression.

Simplifying the Tangent Term

We know that $\tan (x - 180^{\circ}) = -\tan x$. We can substitute this value into the expression.

Simplifying the Cosine Term

We know that $\cos (360^{\circ} + x) = \cos x$. We can substitute this value into the expression.

Substituting the Simplified Terms

After simplifying the sine, tangent, and cosine terms, we get:

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

Simplifying the Expression Further

We can simplify the expression further by combining the terms.

Applying the Pythagorean Identity

We know that $\sin ^2 x + \cos ^2 x = 1$. We can use this identity to simplify the expression.

Simplifying the Expression using the Pythagorean Identity

After applying the Pythagorean identity, we get:

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

Simplifying the Expression Further

We can simplify the expression further by combining the terms.

Final Simplification

After simplifying the expression further, we get:

$${ \frac{-\sin x \cdot \tan x \cdot \cos x}{2\sin ^2 x} }$$

Final Answer

The final answer is:

$${ \frac{-\tan x \cdot \cos x}{2} }$$

Q&A

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

A: The final answer is $\frac{-\tan x \cdot \cos x}{2}$.

Q: How do we simplify the expression using trigonometric identities?

A: We can simplify the expression using trigonometric identities such as $\sin (180^{\circ} - x) = \sin x$, $\sin (180^{\circ} + x) = -\sin x$, $\tan (x - 180^{\circ}) = -\tan x$, and $\cos (360^{\circ} + x) = \cos x$.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is $\sin ^2 x + \cos ^2 x = 1$.

Q: How do we apply the Pythagorean identity to simplify the expression?

A: We can apply the Pythagorean identity by substituting $\sin ^2 x + \cos ^2 x = 1$ into the expression.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{-\tan x \cdot \cos x}{2}$.

Q: What is the main concept of this article?

A: The main concept of this article is to simplify the given expression using trigonometric identities and formulas.

Q: What are the key steps to simplify the expression?

A: The key steps to simplify the expression are:

• Simplify the sine terms using trigonometric identities
• Simplify the tangent term using trigonometric identities
• Simplify the cosine term using trigonometric identities
• Substitute the simplified terms into the expression
• Simplify the expression further using the Pythagorean identity
• Final simplification

Q: What is the importance of simplifying the expression?

A: Simplifying the expression is important because it helps us to understand the underlying concepts and relationships between the trigonometric functions. It also helps us to apply the concepts to real-world problems and applications.