Problem 85 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\frac{\sin x+\sin 5 x}{\cos x+\cos 5 x}=\tan 3 x$$

Step-by-Step Solution

Answer

To prove the identity, simplify using sum-to-product identities and cancellation, resulting in \(\tan(3x)\).

Step 1: Use the sum-to-product identities

The first step is to apply the sum-to-product identities for sine and cosine. For sine, we use the identity:\[sin A + sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\]For cosine, we use:\[cos A + cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\]Substitute these identities into the given expression:\[\frac{2 \sin\left(\frac{6x}{2}\right) \cos\left(\frac{-4x}{2}\right)}{2 \cos\left(\frac{6x}{2}\right) \cos\left(\frac{4x}{2}\right)}\]

Step 2: Simplify the fractions

Since both the numerator and the denominator have a factor of 2, we can cancel it out:\[\frac{\sin(3x) \cos(-2x)}{\cos(3x) \cos(2x)}\]Since \(\cos(-\theta) = \cos(\theta)\), we get:\[\frac{\sin(3x) \cos(2x)}{\cos(3x) \cos(2x)}\]

Step 3: Further simplify the expression

Since \(\cos(2x)\) appears in both the numerator and the denominator, you can cancel it out, assuming \(\cos(2x) eq 0\):\[\frac{\sin(3x)}{\cos(3x)}\]

Step 4: Compare with tangent function

Recognize that the remaining expression is the tangent of \(3x\):\[\frac{\sin(3x)}{\cos(3x)} = \tan(3x)\]This completes the proof of the identity.

Key Concepts

Sum-to-Product Identities

Trigonometric identities are formulas involving trigonometric functions that can be used to transform expressions into simpler or more useful forms. One of these groups of identities is the sum-to-product identities, which are useful when dealing with sums of sine or cosine functions. These identities rewrite the sum of two trigonometric functions as a product.

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

By applying these identities, we can transform a complex expression into a simpler product form. This technique can help make solving trigonometric equations or proving identities easier. In our example, these identities helped us rewrite the original fraction and simplify it effectively.

Tangent Function

The tangent function is one of the primary trigonometric functions and is defined as the ratio of the sine and cosine of an angle:
\[ \tan(x) = \frac{\sin(x)}{\cos(x)} \]When working with trigonometric identities, recognizing expressions that resemble this fundamental definition is key to simplification. In the exercise, simplifying the fraction in the identity led directly to this form,
showcasing how the properties of tangent can be leveraged for verification purposes. By expressing \( \tan(3x) \) as \( \frac{\sin(3x)}{\cos(3x)} \), we were able to recognize that the simplified expression matched the target identity, thus proving the original statement.

Trigonometric Simplification

Trigonometric simplification involves reducing trigonometric expressions to a simpler or more convenient form. This often involves several strategies, such as canceling common factors, using known identities, and recognizing patterns.

• Cancel common terms like coefficients or factors that appear both in the numerator and denominator.
• Apply trigonometric identities such as Pythagorean, angle sum, or sum-to-product identities to transform or simplify expressions.

In our exercise, we utilized these strategies by transforming sums to products using sum-to-product identities and then simplifying by canceling common trigonometric terms. This process demonstrates how a seemingly complicated trigonometric equation can be made straightforward by systematically applying these techniques.