Verify The Identity:$\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos X - \sin X$\]

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Introduction

In trigonometry, verifying identities is a crucial skill that helps us simplify complex expressions and solve problems more efficiently. In this article, we will focus on verifying the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$. This identity involves the cosine function and its properties, and it is essential to understand how to manipulate trigonometric expressions to prove its validity.

The Angle Addition Formula

The angle addition formula for cosine states that $\cos (A + B) = \cos A \cos B - \sin A \sin B$. We can use this formula to expand the left-hand side of the given identity.

$\cos \left(x+\frac{\pi}{4}\right) = \cos x \cos \frac{\pi}{4} - \sin x \sin \frac{\pi}{4}$

Using the values of cosine and sine of $\frac{\pi}{4}$, which are $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$ respectively, we can simplify the expression.

$\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \cos x - \frac{\sqrt{2}}{2} \sin x$

Simplifying the Expression

Now, we can see that the expression on the right-hand side of the identity is similar to the expression we obtained in the previous step. However, we need to manipulate the expression to match the given identity.

$\frac{\sqrt{2}}{2} \cos x - \frac{\sqrt{2}}{2} \sin x = \frac{\sqrt{2}}{2}(\cos x - \sin x)$

This simplification is possible because we can factor out the common term $\frac{\sqrt{2}}{2}$ from both terms.

Conclusion

In this article, we verified the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$. We used the angle addition formula for cosine and simplified the expression to match the given identity. This identity is essential in trigonometry, and understanding how to verify it will help us solve problems more efficiently.

Applications of the Identity

The identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$ has several applications in trigonometry and other areas of mathematics. For example, it can be used to simplify complex trigonometric expressions and solve problems involving right triangles.

Example Problems

Here are a few example problems that involve the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$.

Problem 1

Simplify the expression $\cos \left(x+\frac{\pi}{4}\right) + \sin \left(x+\frac{\pi}{4}\right)$.

Solution

Using the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$, we can simplify the expression as follows:

$\cos \left(x+\frac{\pi}{4}\right) + \sin \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x) + \frac{\sqrt{2}}{2}(\sin x + \cos x)$

Combining like terms, we get:

$\cos \left(x+\frac{\pi}{4}\right) + \sin \left(x+\frac{\pi}{4}\right) = \sqrt{2} \cos x$

Problem 2

Solve the equation $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$.

Solution

Using the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$, we can rewrite the equation as follows:

$\frac{\sqrt{2}}{2}(\cos x - \sin x) = \frac{\sqrt{2}}{2}$

Dividing both sides by $\frac{\sqrt{2}}{2}$, we get:

$\cos x - \sin x = 1$

Solving for $\cos x$, we get:

$\cos x = 1 + \sin x$

This is a linear equation in $\cos x$, and we can solve it using standard methods.

Problem 3

Simplify the expression $\sin \left(x+\frac{\pi}{4}\right) + \cos \left(x+\frac{\pi}{4}\right)$.

Solution

Using the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$, we can simplify the expression as follows:

$\sin \left(x+\frac{\pi}{4}\right) + \cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\sin x + \cos x) + \frac{\sqrt{2}}{2}(\cos x - \sin x)$

Combining like terms, we get:

$\sin \left(x+\frac{\pi}{4}\right) + \cos \left(x+\frac{\pi}{4}\right) = \sqrt{2} \cos x$

Conclusion

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Frequently Asked Questions

Q: What is the angle addition formula for cosine?

A: The angle addition formula for cosine states that $\cos (A + B) = \cos A \cos B - \sin A \sin B$.

Q: How do I verify the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$?

A: To verify the identity, you can use the angle addition formula for cosine and simplify the expression. Start by expanding the left-hand side of the identity using the angle addition formula, and then simplify the expression to match the given identity.

Q: What are some common applications of the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$?

A: The identity has several applications in trigonometry and other areas of mathematics. For example, it can be used to simplify complex trigonometric expressions and solve problems involving right triangles.

Q: How do I simplify the expression $\cos \left(x+\frac{\pi}{4}\right) + \sin \left(x+\frac{\pi}{4}\right)$?

A: To simplify the expression, you can use the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$ and then combine like terms.

Q: How do I solve the equation $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$?

A: To solve the equation, you can use the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$ and then solve for $\cos x$.

Q: What are some common mistakes to avoid when verifying the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$?

A: Some common mistakes to avoid include:

• Not using the angle addition formula for cosine
• Not simplifying the expression correctly
• Not combining like terms
• Not solving for the correct variable

Q: How do I know if the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$ is true?

A: To verify the identity, you can use the angle addition formula for cosine and simplify the expression. If the simplified expression matches the given identity, then the identity is true.

Q: What are some real-world applications of the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$?

A: The identity has several real-world applications, including:

• Simplifying complex trigonometric expressions in physics and engineering
• Solving problems involving right triangles in architecture and construction
• Modeling periodic phenomena in biology and chemistry

Conclusion

In this article, we answered some frequently asked questions about verifying the identity $\cos \left(x+\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\cos x - \sin x)$. We covered topics such as the angle addition formula for cosine, simplifying expressions, solving equations, and common mistakes to avoid. We also discussed real-world applications of the identity and how it can be used to simplify complex trigonometric expressions and solve problems involving right triangles.