Problem 17 - Precalculus: Functions and Graphs

Question

Verify the identity. $$\frac{\sin 4 t+\sin 6 t}{\cos 4 t-\cos 6 t}=\cot t$$

Step-by-Step Solution

Answer

The identity is verified: \(\frac{\sin 4 t+\sin 6 t}{\cos 4 t-\cos 6 t} = \cot t \).

Step 1: Use Sum-to-Product Formula

Start by applying the sum-to-product identities for sine and cosine to simplify the given expression on the left side of the equation.For the numerator, use:\[ \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \]So for \( \sin 4t + \sin 6t \):\[ 2 \sin \left( \frac{4t + 6t}{2} \right) \cos \left( \frac{4t - 6t}{2} \right) = 2 \sin 5t \cos (-t) = 2 \sin 5t \cos t \]For the denominator, use:\[ \cos A - \cos B = -2 \sin \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right) \]So for \( \cos 4t - \cos 6t \):\[ -2 \sin \left( \frac{4t + 6t}{2} \right) \sin \left( \frac{4t - 6t}{2} \right) = -2 \sin 5t \sin (-t) = 2 \sin 5t \sin t \]

Step 2: Simplify the Expression

Substitute the transformed expressions from Step 1 back into the original equation:\[ \frac{2 \sin 5t \cos t}{2 \sin 5t \sin t} \]The \(2 \sin 5t\) terms in the numerator and the denominator cancel each other out.So we are left with:\[ \frac{\cos t}{\sin t} \]

Step 3: Recognize the Cotangent

Recall that the cotangent function is defined as the cosine divided by the sine:\[ \cot t = \frac{\cos t}{\sin t} \]Therefore, we find that:\[ \frac{\cos t}{\sin t} = \cot t \]

Step 4: Conclusion: Confirming the Identity

As we see from Step 3, the left side \(\frac{\sin 4 t+\sin 6 t}{\cos 4 t-\cos 6 t}\) simplifies to \(\cot t\), which is exactly the right side of the equation. Hence, the identity is verified.

Key Concepts

Understanding Sum-to-Product Formulas

Sum-to-product formulas are useful tools in trigonometry that allow you to convert the sum or difference of trigonometric functions into a product of functions. This transformation is advantageous as it simplifies complex expressions and makes them easier to work with.

In our exercise, we used the sum-to-product formula for sine:

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

and for cosine:

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

By applying these identities to the constants given, the original terms of \( \sin 4t + \sin 6t \) and \( \cos 4t - \cos 6t \) were transformed into simpler forms. These simplified expressions make further calculations less daunting, paving the way for a more straightforward approach to verifying the identity.

Exploring the Cotangent Function

The cotangent function, abbreviated as \( \cot \), is one of the fundamental trigonometric functions. It's closely related to the tangent function, but while the tangent is the ratio of the sine to the cosine, the cotangent is the reciprocal of this relationship.

In mathematical terms, it can be expressed as:

• \( \cot t = \frac{\cos t}{\sin t} \)

In our exercise, recognizing that the transformation sequence brought the expression to \( \frac{\cos t}{\sin t} \), it immediately hinted that the result was \( \cot t \). Understanding these basic trigonometric identities ensures we can easily identify such simplifications or transformations, especially when working with equations that require verification.

Mastering Trigonometric Simplification

Trigonometric simplification is the process of rewriting trigonometric expressions in a simpler or more standard form. This can be critical in solving equations, verifying identities, or even in calculus and real-world applications.

In the final step of our solution, we simplified the expression \( \frac{2 \sin 5t \cos t}{2 \sin 5t \sin t} \) by canceling out the common terms, resulting in \( \frac{\cos t}{\sin t} \). Such a simplification process illustrates the power of recognizing patterns and identities in trigonometry.

Simplification not only helps in confirming identities but also minimizes the potential errors while handling more complex trigonometric expressions. Regular practice and familiarity with a range of identities and properties, such as the Pythagorean identities, co-function identities, and the ones used in this exercise, are key to mastering trigonometric simplification. This ensures you handle trigonometric problems with confidence and clarity.