Problem 71 - Algebra and Trigonometry

Question

59–76 Prove the identity. $$\frac{\sin x+\sin 5 x}{\cos x+\cos 5 x}=\tan 3 x$$

Step-by-Step Solution

Answer

The identity is proved: \(\frac{\sin x + \sin 5x}{\cos x + \cos 5x} = \tan 3x\).

Step 1: Use Sum-to-Product Identities

Start with the identity for the numerator \(\sin a + \sin b = 2 \sin(\frac{a+b}{2}) \cos(\frac{a-b}{2})\) and the denominator \(\cos a + \cos b = 2 \cos(\frac{a+b}{2}) \cos(\frac{a-b}{2})\). Set \(a = x\) and \(b = 5x\).

Step 2: Simplify the Numerator

For the numerator, apply the sum-to-product formula: \(\sin x + \sin 5x = 2 \sin(\frac{x+5x}{2}) \cos(\frac{x-5x}{2}) = 2 \sin(3x) \cos(2x)\).

Step 3: Simplify the Denominator

For the denominator, use the sum-to-product formula: \(\cos x + \cos 5x = 2 \cos(\frac{x+5x}{2}) \cos(\frac{x-5x}{2}) = 2 \cos(3x) \cos(2x)\).

Step 4: Form the Expression

Now, express the original function: \(\frac{\sin x + \sin 5x}{\cos x + \cos 5x} = \frac{2 \sin(3x) \cos(2x)}{2 \cos(3x) \cos(2x)}\).

Step 5: Simplify the Expression

Cancel out the common terms of \(2\) and \(\cos(2x)\) from both the numerator and the denominator: \(\frac{\sin(3x) \cos(2x)}{\cos(3x) \cos(2x)} = \frac{\sin(3x)}{\cos(3x)}\).

Step 6: Conclude the Identity

Recognize that \(\frac{\sin(3x)}{\cos(3x)} = \tan(3x)\). Thus, \(\frac{\sin x + \sin 5x}{\cos x + \cos 5x} = \tan 3x\) is proved.

Key Concepts

Sum-to-Product Identities

Trigonometric identities are essential tools in simplifying expressions and proving equations. One particularly useful set are the Sum-to-Product Identities. These formulae help in rewriting sums of sine and cosine functions into products, which are often simpler to handle.
Specifically, the Sum-to-Product Identity for sine is given by:

• \[ \sin a + \sin b = 2 \sin\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \]

This identity allows us to transform two individual sine terms into a single sine function multiplied by a cosine function. This transformation is particularly useful when dealing with trigonometric expressions in fraction form, allowing easy simplification.
Similarly, the identity for cosine is:

• \[ \cos a + \cos b = 2 \cos\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \]

Analogous to sine, this converts the sum of cosines into a product of cosines, which simplifies complex expressions and facilitates cancellation of terms, as demonstrated in the original exercise. Recognizing and applying these identities effectively is key in simplifying trigonometric expressions and proving identities.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a fundamental skill in solving complex trigonometric identities and equations. The primary goal in simplification is to reduce the expression to its simplest form by combining like terms, cancelling terms, or applying identities.
In the given exercise, the expression \(\frac{\sin x + \sin 5x}{\cos x + \cos 5x}\) is simplified using Sum-to-Product Identities for both the numerator and denominator.
Here's how simplification occurs:

• The sum-to-product identity is applied to change the sums of sine and cosine into products:
• For the numerator: \[ \sin x + \sin 5x = 2 \sin(3x) \cos(2x) \]
• For the denominator: \[ \cos x + \cos 5x = 2 \cos(3x) \cos(2x) \]
• This transformation simplifies the fractional expression, allowing the product terms \(2\) and \(\cos(2x)\) to be cancelled easily.

After cancellation, the expression reduces to \(\frac{\sin(3x)}{\cos(3x)}\), which is a simpler form. This process underscores how strategic use of identities and cancellation can lead to concise and manageable trigonometric solutions.

Proving Trigonometric Identities

Proving trigonometric identities involves demonstrating that two different-looking trigonometric expressions are actually equivalent. This process requires strategic manipulation and application of known identities.
In our exercise, the goal is to prove that \(\frac{\sin x + \sin 5x}{\cos x + \cos 5x} = \tan 3x\).
Here's a step-by-step approach to proving this:

• Identify applicable identities or transformations. In this scenario, we use Sum-to-Product Identities to transform the numerator and denominator into simpler expressions.
• Simplify the fraction using these identities to arrive at an expression that can be easily manipulated or reduced.
• In this exercise, simplifying \(\frac{\sin x + \sin 5x}{\cos x + \cos 5x}\) enables cancellation of common terms, resulting in \(\frac{\sin(3x)}{\cos(3x)}\).
• Recognize the resulting form as the tangent function: \(\tan(3x) = \frac{\sin(3x)}{\cos(3x)}\).

Thus, the original expression is proven to be equal to \(\tan 3x\). Approaching proofs in a systematic way, by breaking down complex parts and applying relevant identities, are key strategies in mastering trigonometric proofs.