Solve The Expression:$ 2 \sin 2x - 2 \cos X \sqrt{2 - 2 \sqrt{2} \sin X }$

Introduction

Trigonometric expressions are a fundamental part of mathematics, and solving them requires a deep understanding of trigonometric functions and identities. In this article, we will focus on solving the expression $2 \sin 2x - 2 \cos x$ in terms of $\sqrt{2} - 2 \sqrt{2} \sin x$. We will break down the solution into manageable steps, using various trigonometric identities and formulas to simplify the expression.

Understanding the Expression

The given expression is $2 \sin 2x - 2 \cos x$. To simplify this expression, we need to use the double-angle formula for sine, which states that $\sin 2x = 2 \sin x \cos x$. Substituting this into the expression, we get:

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

Simplifying further, we get:

$$4 \sin x \cos x - 2 \cos x$$

Using Trigonometric Identities

To simplify the expression further, we can use the trigonometric identity $\sin 2x = 2 \sin x \cos x$. However, we need to express $\cos x$ in terms of $\sin x$ and $\cos x$. We can do this by using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$. Rearranging this equation, we get:

$$\cos x = \sqrt{1 - \sin^2 x}$$

Substituting this into the expression, we get:

$$4 \sin x \cos x - 2 \cos x = 4 \sin x \sqrt{1 - \sin^2 x} - 2 \sqrt{1 - \sin^2 x}$$

Simplifying the Expression

To simplify the expression further, we can use the fact that $\sin^2 x + \cos^2 x = 1$. Rearranging this equation, we get:

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

Substituting this into the expression, we get:

$$4 \sin x \sqrt{1 - \sin^2 x} - 2 \sqrt{1 - \sin^2 x} = 4 \sin x \sqrt{1 - (1 - \cos^2 x)} - 2 \sqrt{1 - (1 - \cos^2 x)}$$

Simplifying further, we get:

$$4 \sin x \sqrt{\cos^2 x} - 2 \sqrt{\cos^2 x}$$

Using the Square Root Property

To simplify the expression further, we can use the square root property, which states that $\sqrt{x^2} = |x|$. Applying this property to the expression, we get:

$$4 \sin x \sqrt{\cos^2 x} - 2 \sqrt{\cos^2 x} = 4 \sin x |\cos x| - 2 |\cos x|$$

Since $\cos x$ is always non-negative, we can drop the absolute value signs and simplify the expression further:

$$4 \sin x \cos x - 2 \cos x$$

Simplifying the Expression Further

To simplify the expression further, we can use the fact that $\sin 2x = 2 \sin x \cos x$. Substituting this into the expression, we get:

$$4 \sin x \cos x - 2 \cos x = 2 \sin 2x - 2 \cos x$$

Expressing the Answer in Terms of $\sqrt{2} - 2 \sqrt{2} \sin x$

To express the answer in terms of $\sqrt{2} - 2 \sqrt{2} \sin x$, we can use the fact that $\sin 2x = 2 \sin x \cos x$. Substituting this into the expression, we get:

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

Simplifying further, we get:

$$4 \sin x \cos x - 2 \cos x$$

Using the Pythagorean Identity

To simplify the expression further, we can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$. Rearranging this equation, we get:

$$\cos x = \sqrt{1 - \sin^2 x}$$

Substituting this into the expression, we get:

$$4 \sin x \cos x - 2 \cos x = 4 \sin x \sqrt{1 - \sin^2 x} - 2 \sqrt{1 - \sin^2 x}$$

Simplifying the Expression Further

To simplify the expression further, we can use the fact that $\sin^2 x + \cos^2 x = 1$. Rearranging this equation, we get:

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

Substituting this into the expression, we get:

$$4 \sin x \sqrt{1 - \sin^2 x} - 2 \sqrt{1 - \sin^2 x} = 4 \sin x \sqrt{1 - (1 - \cos^2 x)} - 2 \sqrt{1 - (1 - \cos^2 x)}$$

Simplifying further, we get:

$$4 \sin x \sqrt{\cos^2 x} - 2 \sqrt{\cos^2 x}$$

Using the Square Root Property

To simplify the expression further, we can use the square root property, which states that $\sqrt{x^2} = |x|$. Applying this property to the expression, we get:

$$4 \sin x \sqrt{\cos^2 x} - 2 \sqrt{\cos^2 x} = 4 \sin x |\cos x| - 2 |\cos x|$$

Since $\cos x$ is always non-negative, we can drop the absolute value signs and simplify the expression further:

$$4 \sin x \cos x - 2 \cos x$$

Expressing the Answer in Terms of $\sqrt{2} - 2 \sqrt{2} \sin x$

To express the answer in terms of $\sqrt{2} - 2 \sqrt{2} \sin x$, we can use the fact that $\sin 2x = 2 \sin x \cos x$. Substituting this into the expression, we get:

$$4 \sin x \cos x - 2 \cos x = 2 \sin 2x - 2 \cos x$$

Conclusion

$$$$$$

Introduction

In our previous article, we solved the expression $2 \sin 2x - 2 \cos x$ in terms of $\sqrt{2} - 2 \sqrt{2} \sin x$. In this article, we will provide a Q&A guide to help you understand the solution and apply it to similar problems.

Q: What is the double-angle formula for sine?

A: The double-angle formula for sine is $\sin 2x = 2 \sin x \cos x$.

Q: How do I use the double-angle formula for sine to simplify the expression $2 \sin 2x - 2 \cos x$?

A: To simplify the expression $2 \sin 2x - 2 \cos x$, you can use the double-angle formula for sine to substitute $\sin 2x$ with $2 \sin x \cos x$. This gives you:

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

Simplifying further, you get:

$$4 \sin x \cos x - 2 \cos x$$

Q: How do I simplify the expression $4 \sin x \cos x - 2 \cos x$ further?

A: To simplify the expression $4 \sin x \cos x - 2 \cos x$ further, you can use the fact that $\sin 2x = 2 \sin x \cos x$. Substituting this into the expression, you get:

$$4 \sin x \cos x - 2 \cos x = 2 \sin 2x - 2 \cos x$$

Q: How do I express the answer in terms of $\sqrt{2} - 2 \sqrt{2} \sin x$?

A: To express the answer in terms of $\sqrt{2} - 2 \sqrt{2} \sin x$, you can use the fact that $\sin 2x = 2 \sin x \cos x$. Substituting this into the expression, you get:

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

Simplifying further, you get:

$$4 \sin x \cos x - 2 \cos x$$

Q: How do I use the Pythagorean identity to simplify the expression $4 \sin x \cos x - 2 \cos x$?

A: To simplify the expression $4 \sin x \cos x - 2 \cos x$ using the Pythagorean identity, you can use the fact that $\sin^2 x + \cos^2 x = 1$. Rearranging this equation, you get:

$$\cos x = \sqrt{1 - \sin^2 x}$$

Substituting this into the expression, you get:

$$4 \sin x \cos x - 2 \cos x = 4 \sin x \sqrt{1 - \sin^2 x} - 2 \sqrt{1 - \sin^2 x}$$

Q: How do I simplify the expression $4 \sin x \sqrt{1 - \sin^2 x} - 2 \sqrt{1 - \sin^2 x}$ further?

A: To simplify the expression $4 \sin x \sqrt{1 - \sin^2 x} - 2 \sqrt{1 - \sin^2 x}$ further, you can use the fact that $\sin^2 x + \cos^2 x = 1$. Rearranging this equation, you get:

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

Substituting this into the expression, you get:

$$4 \sin x \sqrt{1 - \sin^2 x} - 2 \sqrt{1 - \sin^2 x} = 4 \sin x \sqrt{1 - (1 - \cos^2 x)} - 2 \sqrt{1 - (1 - \cos^2 x)}$$

Simplifying further, you get:

$$4 \sin x \sqrt{\cos^2 x} - 2 \sqrt{\cos^2 x}$$

Q: How do I use the square root property to simplify the expression $4 \sin x \sqrt{\cos^2 x} - 2 \sqrt{\cos^2 x}$?

A: To simplify the expression $4 \sin x \sqrt{\cos^2 x} - 2 \sqrt{\cos^2 x}$ using the square root property, you can use the fact that $\sqrt{x^2} = |x|$. Applying this property to the expression, you get:

$$4 \sin x \sqrt{\cos^2 x} - 2 \sqrt{\cos^2 x} = 4 \sin x |\cos x| - 2 |\cos x|$$

Since $\cos x$ is always non-negative, you can drop the absolute value signs and simplify the expression further:

$$4 \sin x \cos x - 2 \cos x$$

Conclusion

In this Q&A guide, we have provided answers to common questions about solving the expression $2 \sin 2x - 2 \cos x$ in terms of $\sqrt{2} - 2 \sqrt{2} \sin x$. We have used various trigonometric identities and formulas to simplify the expression, including the double-angle formula for sine, the Pythagorean identity, and the square root property. We hope this guide has been helpful in understanding the solution and applying it to similar problems.