Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity. $$ \frac{\cos t+\cos 4 t+\cos 7 t}{\sin t+\sin 4 t+\sin 7 t}=\cot 4 t $$

Step-by-Step Solution

Answer

The given identity is verified: \( \cot 4t = \frac{\cos t + \cos 4t + \cos 7t}{\sin t + \sin 4t + \sin 7t} \).

Step 1: Break Down the Left Side

First, we notice that the problem presents a fraction with a sum of cosines in the numerator and a sum of sines in the denominator. We need to work on simplifying these trigonometric expressions. We'll use the sum-to-product identities to rewrite these sums.The sum-to-product identities are:- \( \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \)- \( \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \)

Step 2: Apply Sum-to-Product Identities

Apply the identities to simplify the numerator and denominator.For the numerator \( \cos t + \cos 4t + \cos 7t \):- Combine \( \cos t \) and \( \cos 7t \): \[ \cos t + \cos 7t = 2 \cos\left(4t\right) \cos\left(-3t\right) \]- Thus, the numerator becomes: \[ 2 \cos(4t) \cos(3t) + \cos(4t) \]For the denominator \( \sin t + \sin 4t + \sin 7t \):- Combine \( \sin t \) and \( \sin 7t \): \[ \sin t + \sin 7t = 2 \sin(4t) \cos(3t) \]- Thus, the denominator becomes: \[ 2 \sin(4t) \cos(3t) + \sin(4t) \]

Step 3: Simplify Expression

Now, simplify the entire expression: \[ \frac{2 \cos(4t) \cos(3t) + \cos(4t)}{2 \sin(4t) \cos(3t) + \sin(4t)} \]Factor out \( \cos(4t) \) in the numerator and \( \sin(4t) \) in the denominator:- Numerator: \( \cos(4t) (2\cos(3t) + 1) \)- Denominator: \( \sin(4t) (2\cos(3t) + 1) \)

Step 4: Cancel Common Terms

Since the numerator and the denominator both have the common factor \((2\cos(3t) + 1)\), cancel this out:\[ \frac{\cos(4t)}{\sin(4t)} \]which simplifies directly to \( \cot(4t) \).

Step 5: Verify the Identity

After simplification, we are left with:\[ \frac{\cos(4t)}{\sin(4t)} = \cot(4t) \]The left side is equivalent to the right side, thereby verifying the identity.

Key Concepts

Sum-to-Product Identities

Sum-to-product identities help in transforming the sum of trigonometric functions into a single product. This can simplify complicated expressions and make verifying identities much easier.
These identities are particularly handy when dealing with problems involving multiple angles. Here's a quick look at the formulas:

• For cosine: \( \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \)
• For sine: \( \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \)

When working with these identities, always pay attention to the argument angles \( A \) and \( B \) that you are summing.
This makes sure that you correctly find the average and half-difference of the angles. It also simplifies the trigonometric operations involved, which is crucial for verifying complex identities.

Verifying Identities

The process of verifying trigonometric identities involves demonstrating that two different expressions are indeed equivalent.
In our original exercise, this means showing that the left-hand side of the given identity equals the right-hand side, which is \( \cot 4t \).
Here’s a structured approach to verifying:

• Simplify both sides separately as much as possible.
• Utilize known identities, such as sum-to-product identities, to rework complex expressions.
• Look for common factors that appear in both the numerator and denominator, allowing for cancellation.

If after all these steps, both sides match, the identity has been verified.
This process not only proves equivalency but also reinforces understanding of trigonometric functions and their properties.

Trigonometric Simplification

Trigonometric simplification is a powerful tool that helps in reducing complex trigonometric expressions into simpler forms.
This process can reveal hidden patterns or factors that might not be obvious at first glance. In our example, once new expressions are formed using sum-to-product identities, simplification can be pursued further by:

• Factoring out common terms.
• Cancelling similar terms in numerators and denominators.
• Employing basic trigonometric identities until achieving a definitive expression, like \( \cot 4t \).

The goal is to ensure that all parts of the expression have been thoroughly simplified.
By doing so, students gain an appreciation of the symmetry and elegance that often lurk within trigonometric identities.