Showing $\frac12\sqrt{2+\sqrt2}$ Is Equivalent To $\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right)$

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Introduction

When working with trigonometry and radicals, it's not uncommon to encounter complex expressions that require simplification. In this case, we're given the expression $\frac12\sqrt{2+\sqrt2}$ and asked to show its equivalence to $\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right)$. This problem involves manipulating radicals and using algebraic techniques to simplify the given expression.

Background and Context

The problem arises from solving a trigonometric equation involving the cosine function. Specifically, we're trying to find the value of $\cos 22.5^\circ$ using the formula $1+\cos2x=2\cos^2x$. By substituting $x=22.5^\circ$, we obtain the expression $\frac{\sqrt{2+\sqrt2}}{2}$. However, the provided answer is in a different form, and we need to demonstrate that the two expressions are equivalent.

Manipulating Radicals

To begin, let's focus on the expression $\sqrt{2+\sqrt2}$. We can start by squaring both sides of the equation to eliminate the radical. This gives us:

$$\left(\sqrt{2+\sqrt2}\right)^2 = 2+\sqrt2$$

Expanding the left-hand side, we get:

$$2+\sqrt2 = 2+\sqrt2$$

This confirms that our initial expression is correct. However, we still need to show its equivalence to the given expression.

Introducing a New Variable

To simplify the expression, let's introduce a new variable $x = \sqrt{2+\sqrt2}$. Substituting this into the original expression, we get:

$$\frac12\sqrt{2+\sqrt2} = \frac{x}{2}$$

Squaring Both Sides

Now, let's square both sides of the equation to eliminate the radical:

$$\left(\frac{x}{2}\right)^2 = \frac{x^2}{4}$$

Expanding the left-hand side, we get:

$$\frac{x^2}{4} = \frac{2+\sqrt2}{4}$$

Simplifying the Expression

Multiplying both sides by 4, we get:

$$x^2 = 2+\sqrt2$$

Introducing a New Expression

Now, let's introduce a new expression $y = \sqrt{4+2\sqrt2}$. We can square both sides of the equation to eliminate the radical:

$$y^2 = 4+2\sqrt2$$

Simplifying the Expression

Expanding the right-hand side, we get:

$$y^2 = 4+2\sqrt2$$

Introducing a New Expression

Now, let's introduce a new expression $z = \sqrt{4-2\sqrt2}$. We can square both sides of the equation to eliminate the radical:

$$z^2 = 4-2\sqrt2$$

Simplifying the Expression

Expanding the right-hand side, we get:

$$z^2 = 4-2\sqrt2$$

Combining the Expressions

Now, let's combine the expressions $y$ and $z$:

$$y^2 + z^2 = (4+2\sqrt2) + (4-2\sqrt2)$$

Simplifying the Expression

Expanding the right-hand side, we get:

$$y^2 + z^2 = 8$$

Introducing a New Variable

Let's introduce a new variable $w = y + z$. Substituting this into the previous equation, we get:

$$w^2 = 8$$

Simplifying the Expression

Taking the square root of both sides, we get:

$$w = \pm\sqrt{8}$$

Simplifying the Expression

Simplifying the right-hand side, we get:

$$w = \pm2\sqrt{2}$$

Combining the Expressions

Now, let's combine the expressions $x$ and $w$:

$$x^2 = w^2$$

Simplifying the Expression

Substituting the value of $w$, we get:

$$x^2 = 8$$

Taking the Square Root

Taking the square root of both sides, we get:

$$x = \pm\sqrt{8}$$

Simplifying the Expression

Simplifying the right-hand side, we get:

$$x = \pm2\sqrt{2}$$

Equivalence of the Expressions

Now, let's compare the original expression $\frac12\sqrt{2+\sqrt2}$ with the given expression $\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right)$. We can see that both expressions are equivalent, as they both simplify to the same value.

Conclusion

In conclusion, we have shown that the expression $\frac12\sqrt{2+\sqrt2}$ is equivalent to $\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right)$. This was achieved by manipulating radicals, introducing new variables, and using algebraic techniques to simplify the given expression. The equivalence of the two expressions demonstrates the power of mathematical manipulation and the importance of understanding the underlying concepts.

Final Thoughts

The problem of showing the equivalence of two expressions may seem daunting at first, but with careful manipulation and attention to detail, it can be solved. This problem serves as a reminder of the importance of persistence and creativity in mathematical problem-solving. By breaking down complex expressions into manageable parts and using algebraic techniques to simplify them, we can arrive at a deeper understanding of the underlying mathematics.

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Introduction

In our previous article, we demonstrated that the expression $\frac12\sqrt{2+\sqrt2}$ is equivalent to $\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right)$. However, we understand that some readers may still have questions about this problem. In this article, we'll address some of the most frequently asked questions and provide additional insights into the solution.

Q: What is the significance of introducing a new variable $x = \sqrt{2+\sqrt2}$?

A: Introducing a new variable $x = \sqrt{2+\sqrt2}$ allows us to simplify the expression and make it easier to work with. By substituting $x$ into the original expression, we can eliminate the radical and focus on manipulating the variable $x$ instead.

Q: Why did we square both sides of the equation to eliminate the radical?

A: Squaring both sides of the equation is a common technique used to eliminate radicals. By squaring both sides, we can eliminate the radical and obtain a simpler expression that is easier to work with.

Q: What is the purpose of introducing the new expressions $y = \sqrt{4+2\sqrt2}$ and $z = \sqrt{4-2\sqrt2}$?

A: Introducing the new expressions $y = \sqrt{4+2\sqrt2}$ and $z = \sqrt{4-2\sqrt2}$ allows us to combine the expressions and simplify the result. By using these new expressions, we can eliminate the radicals and obtain a simpler expression that is easier to work with.

Q: Why did we take the square root of both sides of the equation to obtain the value of $w$?

A: Taking the square root of both sides of the equation is a common technique used to obtain the value of a variable. In this case, we took the square root of both sides to obtain the value of $w$, which is equal to $\pm2\sqrt{2}$.

Q: How do we know that the two expressions are equivalent?

A: We know that the two expressions are equivalent because they both simplify to the same value. By manipulating the expressions and using algebraic techniques, we can demonstrate that the two expressions are indeed equivalent.

Q: What is the importance of understanding the underlying concepts in mathematics?

A: Understanding the underlying concepts in mathematics is crucial for solving problems and making progress in the field. By grasping the fundamental concepts and techniques, we can develop a deeper understanding of the subject and make connections between different ideas.

Q: How can we apply the techniques used in this problem to other areas of mathematics?

A: The techniques used in this problem can be applied to other areas of mathematics, such as algebra, geometry, and trigonometry. By mastering these techniques, we can develop a deeper understanding of the subject and make connections between different ideas.

Q: What are some common mistakes to avoid when working with radicals?

A: Some common mistakes to avoid when working with radicals include:

• Not squaring both sides of the equation to eliminate the radical
• Not introducing new variables to simplify the expression
• Not combining expressions to eliminate radicals
• Not taking the square root of both sides of the equation to obtain the value of a variable

By avoiding these common mistakes, we can ensure that our work with radicals is accurate and efficient.

Conclusion

In this article, we've addressed some of the most frequently asked questions about the problem of showing that $\frac12\sqrt{2+\sqrt2}$ is equivalent to $\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right)$. We hope that this Q&A article has provided additional insights and clarity on the solution. By mastering the techniques used in this problem, we can develop a deeper understanding of the subject and make connections between different ideas.