Simplify The Expression:$\sin ((n+1)x) \sin ((n+2)x) + \cos ((n+1)x) \cos ((n+2)x) = \cos X$

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Introduction

In this article, we will simplify the given trigonometric expression involving sine and cosine functions. The expression is $\sin ((n+1)x) \sin ((n+2)x) + \cos ((n+1)x) \cos ((n+2)x) = \cos x$. We will use various trigonometric identities and formulas to simplify the expression and arrive at the final result.

Trigonometric Identities

Before we proceed with the simplification, let's recall some basic trigonometric identities that we will use in this article.

• Sine and Cosine Product Formula: $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$
• Cosine Product Formula: $\cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]$
• Cosine Double Angle Formula: $\cos 2A = 2\cos^2 A - 1$
• Sine Double Angle Formula: $\sin 2A = 2\sin A \cos A$

Simplifying the Expression

Now, let's simplify the given expression using the trigonometric identities mentioned above.

$\sin ((n+1)x) \sin ((n+2)x) + \cos ((n+1)x) \cos ((n+2)x)$

Using the Sine and Cosine Product Formula, we can rewrite the expression as:

$\frac{1}{2}[\cos((n+1)x - (n+2)x) - \cos((n+1)x + (n+2)x)] + \frac{1}{2}[\cos((n+1)x - (n+2)x) + \cos((n+1)x + (n+2)x)]$

Simplifying the expression further, we get:

$\cos x$

Derivation of the Expression

Let's derive the given expression using the Cosine Double Angle Formula.

$\cos 2A = 2\cos^2 A - 1$

We can rewrite the expression as:

$\cos 2A = 2\cos^2 A - 1$

$\cos 2A + 1 = 2\cos^2 A$

$\frac{\cos 2A + 1}{2} = \cos^2 A$

$\cos^2 A = \frac{\cos 2A + 1}{2}$

Now, let's substitute $A = (n+1)x$ and $B = (n+2)x$ in the above expression.

$\cos^2 ((n+1)x) = \frac{\cos 2((n+1)x) + 1}{2}$

$\cos^2 ((n+1)x) = \frac{\cos 2(n+1)x + 1}{2}$

$\cos^2 ((n+1)x) = \frac{\cos 2nx + 2\cos 2x + 1}{2}$

$\cos^2 ((n+1)x) = \frac{\cos 2nx + 2\cos 2x + 1}{2}$

Conclusion

In this article, we simplified the given trigonometric expression involving sine and cosine functions. We used various trigonometric identities and formulas to simplify the expression and arrive at the final result. The expression $\sin ((n+1)x) \sin ((n+2)x) + \cos ((n+1)x) \cos ((n+2)x) = \cos x$ can be simplified using the Sine and Cosine Product Formula and the Cosine Double Angle Formula.

Final Answer

The final answer is $\boxed{\cos x}$.

References

• [1] Trigonometry by Michael Corral
• [2] Calculus by Michael Spivak
• [3] Mathematics by G.H. Hardy

Related Articles

• Simplify the Expression: $\sin^2 x + \cos^2 x = 1$
• Derive the Expression: $\cos 2A = 2\cos^2 A - 1$
• Simplify the Expression: $\sin (A+B) = \sin A \cos B + \cos A \sin B$
  

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Frequently Asked Questions

Q: What is the given expression?

A: The given expression is $\sin ((n+1)x) \sin ((n+2)x) + \cos ((n+1)x) \cos ((n+2)x) = \cos x$.

Q: How can we simplify the given expression?

A: We can simplify the given expression using the Sine and Cosine Product Formula and the Cosine Double Angle Formula.

Q: What is the final result of the simplification?

A: The final result of the simplification is $\cos x$.

Q: What are the trigonometric identities used in the simplification?

A: The trigonometric identities used in the simplification are:

• Sine and Cosine Product Formula: $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$
• Cosine Product Formula: $\cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]$
• Cosine Double Angle Formula: $\cos 2A = 2\cos^2 A - 1$
• Sine Double Angle Formula: $\sin 2A = 2\sin A \cos A$

Q: How can we derive the expression $\cos 2A = 2\cos^2 A - 1$?

A: We can derive the expression $\cos 2A = 2\cos^2 A - 1$ using the Cosine Double Angle Formula.

Q: What is the significance of the expression $\cos 2A = 2\cos^2 A - 1$?

A: The expression $\cos 2A = 2\cos^2 A - 1$ is a fundamental trigonometric identity that can be used to simplify various trigonometric expressions.

Q: How can we use the expression $\cos 2A = 2\cos^2 A - 1$ to simplify the given expression?

A: We can use the expression $\cos 2A = 2\cos^2 A - 1$ to simplify the given expression by substituting $A = (n+1)x$ and $B = (n+2)x$.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is $\boxed{\cos x}$.

Related Questions

• Simplify the Expression: $\sin^2 x + \cos^2 x = 1$
• Derive the Expression: $\cos 2A = 2\cos^2 A - 1$
• Simplify the Expression: $\sin (A+B) = \sin A \cos B + \cos A \sin B$

Conclusion

In this article, we answered frequently asked questions related to the simplification of the expression $\sin ((n+1)x) \sin ((n+2)x) + \cos ((n+1)x) \cos ((n+2)x) = \cos x$. We used various trigonometric identities and formulas to simplify the expression and arrive at the final result. The expression $\sin ((n+1)x) \sin ((n+2)x) + \cos ((n+1)x) \cos ((n+2)x) = \cos x$ can be simplified using the Sine and Cosine Product Formula and the Cosine Double Angle Formula.

Final Answer

The final answer is $\boxed{\cos x}$.

References

• [1] Trigonometry by Michael Corral
• [2] Calculus by Michael Spivak
• [3] Mathematics by G.H. Hardy

Related Articles

• Simplify the Expression: $\sin^2 x + \cos^2 x = 1$
• Derive the Expression: $\cos 2A = 2\cos^2 A - 1$
• Simplify the Expression: $\sin (A+B) = \sin A \cos B + \cos A \sin B$