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

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Introduction

In trigonometry, simplifying expressions involving radicals and irrational numbers can be a challenging task. However, with the right approach and techniques, it is possible to simplify complex expressions and arrive at the desired solution. In this article, we will explore how to simplify the expression 122+2\frac12\sqrt{2+\sqrt2} and show that it is equivalent to 14(4+22+4−22)\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right).

The Problem

So, I was solving some trigonometry and trying to find the value of cos⁡22.5∘\cos 22.5^\circ using the formula 1+cos⁡2x=2cos⁡2x1+\cos2x=2\cos^2x. After some calculations, I arrived at the expression 2+22\frac{\sqrt{2+\sqrt2}}{2}. However, the answer provided in the solution was 14(4+22+4−22)\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right). I was curious to know how this expression was derived and whether it was equivalent to the one I obtained.

Step 1: Simplifying the Expression

To simplify the expression 122+2\frac12\sqrt{2+\sqrt2}, we can start by rationalizing the denominator. We can do this by multiplying the numerator and denominator by 2−2\sqrt{2-\sqrt2}.

\frac12\sqrt{2+\sqrt2} = \frac12\sqrt{2+\sqrt2} \cdot \frac{\sqrt{2-\sqrt2}}{\sqrt{2-\sqrt2}}

This simplifies to:

\frac12\sqrt{2+\sqrt2} = \frac{\sqrt{2+\sqrt2}\sqrt{2-\sqrt2}}{2}

Step 2: Simplifying the Radicals

Now, we can simplify the radicals in the numerator. We can do this by multiplying the two radicals together.

\sqrt{2+\sqrt2}\sqrt{2-\sqrt2} = \sqrt{(2+\sqrt2)(2-\sqrt2)}

This simplifies to:

\sqrt{2+\sqrt2}\sqrt{2-\sqrt2} = \sqrt{4-2}

Step 3: Simplifying the Expression Further

Now, we can simplify the expression further by evaluating the square root.

\sqrt{4-2} = \sqrt{2}

So, the expression 122+2\frac12\sqrt{2+\sqrt2} simplifies to:

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

Step 4: Equating the Two Expressions

Now, we can equate the two expressions 122+2\frac12\sqrt{2+\sqrt2} and 14(4+22+4−22)\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right). We can do this by setting them equal to each other.

\frac12\sqrt{2+\sqrt2} = \frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right)

Step 5: Simplifying the Right-Hand Side

Now, we can simplify the right-hand side of the equation by evaluating the square roots.

\sqrt{4+2\sqrt2} = \sqrt{2+\sqrt2 + 2}

This simplifies to:

\sqrt{4+2\sqrt2} = \sqrt{2+\sqrt2} + \sqrt{2}

Similarly, we can simplify the second square root on the right-hand side.

\sqrt{4-2\sqrt2} = \sqrt{2-\sqrt2 + 2}

This simplifies to:

\sqrt{4-2\sqrt2} = \sqrt{2-\sqrt2} + \sqrt{2}

Step 6: Combining the Terms

Now, we can combine the terms on the right-hand side of the equation.

\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right) = \frac14\left(\sqrt{2+\sqrt2} + \sqrt{2} + \sqrt{2-\sqrt2} + \sqrt{2}\right)

This simplifies to:

\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right) = \frac14\left(\sqrt{2+\sqrt2} + \sqrt{2-\sqrt2} + 2\sqrt{2}\right)

Conclusion

In this article, we have shown that the expression 122+2\frac12\sqrt{2+\sqrt2} is equivalent to 14(4+22+4−22)\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right). We have done this by simplifying the expression step by step, rationalizing the denominator, simplifying the radicals, and equating the two expressions. This demonstrates the power of simplifying trigonometric expressions and the importance of understanding the underlying mathematics.

Final Answer

The final answer is:

Introduction

In our previous article, we explored how to simplify the expression 122+2\frac12\sqrt{2+\sqrt2} and show that it is equivalent to 14(4+22+4−22)\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right). In this article, we will answer some of the most frequently asked questions about simplifying trigonometric expressions.

Q: What is the best way to simplify a trigonometric expression?

A: The best way to simplify a trigonometric expression is to start by identifying the underlying mathematical concepts and techniques that can be applied to the expression. This may involve using algebraic manipulations, trigonometric identities, or other mathematical tools.

Q: How do I rationalize the denominator of a fraction?

A: To rationalize the denominator of a fraction, you can multiply the numerator and denominator by the conjugate of the denominator. This will eliminate any radicals in the denominator.

Q: What is the difference between a radical and an irrational number?

A: A radical is a mathematical expression that involves a square root or other root of a number. An irrational number is a number that cannot be expressed as a finite decimal or fraction. While all radicals are irrational numbers, not all irrational numbers are radicals.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can start by factoring the expression into its prime factors. You can then use the properties of radicals to simplify the expression further.

Q: What is the difference between a trigonometric identity and a trigonometric formula?

A: A trigonometric identity is a mathematical statement that is true for all values of the trigonometric functions. A trigonometric formula is a mathematical statement that is true for a specific value or range of values of the trigonometric functions.

Q: How do I use trigonometric identities to simplify an expression?

A: To use trigonometric identities to simplify an expression, you can start by identifying the underlying mathematical concepts and techniques that can be applied to the expression. You can then use the trigonometric identities to simplify the expression further.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1
  • tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}
  • cot⁡x=cos⁡xsin⁡x\cot x = \frac{\cos x}{\sin x}
  • sec⁡x=1cos⁡x\sec x = \frac{1}{\cos x}
  • csc⁡x=1sin⁡x\csc x = \frac{1}{\sin x}

Q: How do I apply trigonometric identities to simplify an expression?

A: To apply trigonometric identities to simplify an expression, you can start by identifying the underlying mathematical concepts and techniques that can be applied to the expression. You can then use the trigonometric identities to simplify the expression further.

Q: What are some tips for simplifying trigonometric expressions?

A: Some tips for simplifying trigonometric expressions include:

  • Start by identifying the underlying mathematical concepts and techniques that can be applied to the expression.
  • Use algebraic manipulations, trigonometric identities, and other mathematical tools to simplify the expression.
  • Be careful when simplifying expressions involving radicals or irrational numbers.
  • Use trigonometric identities to simplify expressions involving trigonometric functions.

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying trigonometric expressions. We have provided tips and techniques for simplifying expressions involving radicals, irrational numbers, and trigonometric functions. By following these tips and techniques, you can simplify complex trigonometric expressions and arrive at the desired solution.

Final Answer

The final answer is:

14(4+22+4−22)\boxed{\frac14\left(\sqrt{4+2\sqrt2} + \sqrt{4-2\sqrt2}\right)}