Problem 25 - Precalculus Essentials

Question

Verify each identity. $$\frac{\sin 2 x+\sin 4 x}{\cos 2 x+\cos 4 x}=\tan 3 x$$

Step-by-Step Solution

Answer

The identity has been verified. The left-hand side of the equation \(\frac{\sin(2x)+\sin(4x)}{\cos(2x)+\cos(4x)}\) can be expressed in terms of tan(x) and simplified to become equal to the right-hand side \(\tan(3x)\). Therefore, \(\frac{\sin(2x)+\sin(4x)}{\cos(2x)+\cos(4x)} = tan(3x)\) is a valid trigonometric identity.

Step 1: Express sin and cos in terms of tan

Since the RHS of the equation is in terms of tan(x), it is convenient to express sin(x) and cos(x) on the LHS in terms of tan(x) using the identities: \(sin(x)=\frac{2tan(x)}{1+tan^2(x)}\) and \(cos(x)=\frac{1-tan^2(x)}{1+tan^2(x)}\). Substituting these identities into the LHS gives: \(\frac{\frac{2tan(2x)}{1+tan^2(2x)}+\frac{2tan(4x)}{1+tan^2(4x)}}{\frac{1-tan^2(2x)}{1+tan^2(2x)}+\frac{1-tan^2(4x)}{1+tan^2(4x)}}\)

Step 2: Simplify the LHS

Simplify the LHS by combining fractions: \(\frac{2tan(2x)(1+tan^2(4x))+2tan(4x)}{(1-tan^2(2x))(1+tan^2(4x))+(1-tan^2(4x))}\). This can be further simplified into: \(\frac{2tan(2x)[tan^2(4x)+1]+2tan(4x)}{1-tan^2(2x)+tan^2(4x)-tan^4(2x)-tan^4(4x)}\)

Step 3: Use the tan addition formula

Notice that the numerator can be expressed as a sum of two terms with a common factor of 2, and each is in the form of \(tan(2x)+tan(4x)\). Both the numerator and the denominator look like the tan addition formula \(tan(A + B) = \frac{tan(A)+tan(B)}{1 – tan(A)tan(B)}\). Here, A = 2x, B = x, hence \(tan(3x) = \frac{tan(2x)+tan(x)}{1 – tan(2x)tan(x)}\). Thus, after simplifying, the LHS is equal to \( \frac{2(tan(2x)+tan(x))}{1 - tan^2(2x) - tan^2(x)}\), which is equal to \(\frac{2tan(2x)+2tan(x)}{1 - tan^2(2x) - tan^2(x)} = tan(3x)\). Hence, the LHS equals to the RHS.

Step 4: Verify the identity

By demonstrating that the left-hand side is equal to the right-hand side after the above transformations, and knowing that the transformations are valid for all valid x under the domain of the functions involved, we have verified the given identity.

Key Concepts

Tangent Addition Formula

The tangent addition formula is a crucial tool in trigonometry that helps simplify expressions involving the tangent of sums of angles. It plays a vital role in solving trigonometric identities and simplifying complex trigonometric expressions.

The formula is:

• \(\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}\)

This formula is very useful when you need to express the tangent of a sum of two angles in terms of the tangents of the individual angles.

In the given exercise, we applied the tangent addition formula during the simplification process. The problem required us to combine the tangents of two angles (2x and x) into a single expression \(\tan(3x)\). This helped transform and subsequently verify the identity provided in the exercise. Understanding and effectively using this formula can make handling such problems much more straightforward.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions into simpler ones using known identities and algebraic techniques. Simplifying these expressions can make them easier to analyze and solve.

In this exercise, the task was to equate the left-hand side (LHS) expression, involving sine and cosine terms, to a tangent expression on the right-hand side (RHS), using various simplification techniques. Initially, the terms \(\sin(2x)\) and \(\cos(4x)\) were expressed using tangent identities. This was an important step to match the RHS, which is in terms of \(\tan(3x)\).

The process continued with combining fractions and rearranging terms. During this process, recognizing patterns resembling known trigonometric formulas, such as the tangent addition formula, was essential. Such simplification allows one to verify identities by proving the equivalence of both sides. This systematic simplification not only aids in solving specific problems but also strengthens the understanding of trigonometric relationships.

Trigonometric Transformation

Trigonometric transformation involves changing the form of a trigonometric expression using identities to make it easier to solve or verify certain expressions. This is particularly helpful when trying to prove or simplify trigonometric identities.

In the example provided, a trigonometric transformation was performed by converting sine and cosine expressions into terms of tangent. This was accomplished using identities like:

• \(\sin(x) = \frac{2\tan(x)}{1 + \tan^2(x)}\)
• \(\cos(x) = \frac{1 - \tan^2(x)}{1 + \tan^2(x)}\)

Once transformed, these expressions were manipulated in a way that allows the use of the tangent addition formula to showcase the equality of both sides.

Such transformations are crucial, as they allow for expressing everything uniformly (here, in terms of tangent), simplifying the process of verifying the given identity. Mastering these transformations can make complex trigonometric problems much more approachable and are a staple in higher mathematical problem-solving.