Prove The Identity:$\[ \cos \left(x-\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{6}-x\right) = \cos X \\]Note That Each Statement Must Be Based On A Rule Chosen From The List Of Rules Provided.

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Introduction

In this article, we will delve into the world of trigonometry and explore a fascinating identity involving cosine and sine functions. The given identity is $\cos \left(x-\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{6}-x\right) = \cos x$. Our goal is to prove this identity using various trigonometric rules and identities.

Background

Before we begin, let's recall some essential trigonometric identities and rules that we will use to prove the given identity. These include:

• Cosine Angle Addition Formula: $\cos (a + b) = \cos a \cos b - \sin a \sin b$
• Sine Angle Addition Formula: $\sin (a + b) = \sin a \cos b + \cos a \sin b$
• Cosine Angle Subtraction Formula: $\cos (a - b) = \cos a \cos b + \sin a \sin b$
• Sine Angle Subtraction Formula: $\sin (a - b) = \sin a \cos b - \cos a \sin b$
• Pythagorean Identity: $\sin^2 x + \cos^2 x = 1$

Step 1: Apply the Cosine Angle Subtraction Formula

We will start by applying the cosine angle subtraction formula to the first term of the given identity: $\cos \left(x-\frac{\pi}{3}\right)$. Using the formula, we get:

$$\cos \left(x-\frac{\pi}{3}\right) = \cos x \cos \frac{\pi}{3} + \sin x \sin \frac{\pi}{3}$$

Step 3: Simplify the Expression

Now, let's simplify the expression by substituting the values of $\cos \frac{\pi}{3}$ and $\sin \frac{\pi}{3}$. We know that $\cos \frac{\pi}{3} = \frac{1}{2}$ and $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$. Substituting these values, we get:

$$\cos \left(x-\frac{\pi}{3}\right) = \cos x \cdot \frac{1}{2} + \sin x \cdot \frac{\sqrt{3}}{2}$$

Step 4: Apply the Sine Angle Subtraction Formula

Next, we will apply the sine angle subtraction formula to the second term of the given identity: $\sin \left(\frac{\pi}{6}-x\right)$. Using the formula, we get:

$$\sin \left(\frac{\pi}{6}-x\right) = \sin \frac{\pi}{6} \cos x - \cos \frac{\pi}{6} \sin x$$

Step 5: Simplify the Expression

Now, let's simplify the expression by substituting the values of $\sin \frac{\pi}{6}$ and $\cos \frac{\pi}{6}$. We know that $\sin \frac{\pi}{6} = \frac{1}{2}$ and $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$. Substituting these values, we get:

$$\sin \left(\frac{\pi}{6}-x\right) = \frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x$$

Step 6: Combine the Terms

Now, let's combine the two terms we obtained in steps 3 and 5:

$$\cos \left(x-\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{6}-x\right) = \cos x \cdot \frac{1}{2} + \sin x \cdot \frac{\sqrt{3}}{2} + \frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x$$

Step 7: Simplify the Expression

Now, let's simplify the expression by combining like terms:

$$\cos \left(x-\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{6}-x\right) = \left(\cos x \cdot \frac{1}{2} + \frac{1}{2} \cos x\right) + \left(\sin x \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} \sin x\right)$$

Step 8: Factor Out Common Terms

Now, let's factor out common terms:

$$\cos \left(x-\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{6}-x\right) = \frac{1}{2} \cos x + \frac{1}{2} \cos x + \frac{\sqrt{3}}{2} \sin x - \frac{\sqrt{3}}{2} \sin x$$

Step 9: Simplify the Expression

Now, let's simplify the expression by combining like terms:

$$\cos \left(x-\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{6}-x\right) = \frac{1}{2} \cos x + \frac{1}{2} \cos x$$

Step 10: Simplify the Expression

Now, let's simplify the expression by combining like terms:

$$\cos \left(x-\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{6}-x\right) = \cos x$$

Conclusion

In this article, we proved the identity $\cos \left(x-\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{6}-x\right) = \cos x$ using various trigonometric rules and identities. We applied the cosine angle subtraction formula, simplified the expression, and factored out common terms to arrive at the final result. This identity is a useful tool in trigonometry and can be used to simplify complex expressions involving cosine and sine functions.

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Introduction

In our previous article, we proved the identity $\cos \left(x-\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{6}-x\right) = \cos x$ using various trigonometric rules and identities. In this article, we will answer some frequently asked questions related to this identity.

Q: What is the significance of this identity?

A: This identity is significant because it shows that the sum of two trigonometric functions, $\cos \left(x-\frac{\pi}{3}\right)$ and $\sin \left(\frac{\pi}{6}-x\right)$, is equal to a single trigonometric function, $\cos x$. This identity can be used to simplify complex expressions involving cosine and sine functions.

Q: How can I use this identity in real-world applications?

A: This identity can be used in various real-world applications, such as:

• Engineering: This identity can be used to simplify complex expressions involving trigonometric functions in engineering problems, such as calculating the stress on a beam or the force on a spring.
• Physics: This identity can be used to simplify complex expressions involving trigonometric functions in physics problems, such as calculating the motion of a pendulum or the trajectory of a projectile.
• Computer Science: This identity can be used to simplify complex expressions involving trigonometric functions in computer science problems, such as calculating the position of a robot or the orientation of a camera.

Q: What are some common mistakes to avoid when proving this identity?

A: Some common mistakes to avoid when proving this identity include:

• Not using the correct trigonometric identities: Make sure to use the correct trigonometric identities, such as the cosine angle subtraction formula and the sine angle subtraction formula.
• Not simplifying the expression correctly: Make sure to simplify the expression correctly by combining like terms and factoring out common terms.
• Not checking the final result: Make sure to check the final result to ensure that it is correct.

Q: Can this identity be used to prove other trigonometric identities?

A: Yes, this identity can be used to prove other trigonometric identities. For example, it can be used to prove the identity $\sin \left(x+\frac{\pi}{4}\right) = \cos x + \sin x$.

Q: How can I apply this identity to solve trigonometric equations?

A: This identity can be used to solve trigonometric equations by simplifying complex expressions involving trigonometric functions. For example, it can be used to solve the equation $\cos \left(x-\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{6}-x\right) = 0$.

Q: What are some other trigonometric identities that are similar to this one?

A: Some other trigonometric identities that are similar to this one include:

• The cosine angle addition formula: $\cos (a + b) = \cos a \cos b - \sin a \sin b$
• The sine angle addition formula: $\sin (a + b) = \sin a \cos b + \cos a \sin b$
• The cosine angle subtraction formula: $\cos (a - b) = \cos a \cos b + \sin a \sin b$
• The sine angle subtraction formula: $\sin (a - b) = \sin a \cos b - \cos a \sin b$

Conclusion

In this article, we answered some frequently asked questions related to the identity $\cos \left(x-\frac{\pi}{3}\right) + \sin \left(\frac{\pi}{6}-x\right) = \cos x$. We discussed the significance of this identity, its real-world applications, common mistakes to avoid, and how to apply it to solve trigonometric equations. We also mentioned some other trigonometric identities that are similar to this one.