Problem 47 - Precalculus with Limits

Question

Exercises 43-52, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. \( \tan^4 2x \)

Step-by-Step Solution

Answer

The expression \( \tan^4 2x \) can be rewritten using power-reducing formulas as \( \frac{1 - 2\cos^2 2x + \cos^4 2x}{\cos^4 2x} \).

Step 1: Expressing in terms of sine and cosine

Let's start by expressing \( \tan \) in terms of sine and cosine. We can rewrite \( \tan^4 2x \) as \( \left( \frac{\sin 2x}{\cos 2x} \right)^4 \).

Step 2: Expanding the Expression

Next we can expand the expression to get \( \frac{\sin^4 2x}{\cos^4 2x} \).

Step 3: Applying the Power-Reducing formula

Now we can use the power-reducing formula for sine and rewrite \( \sin^2 x \) as \( 1 - \cos^2 x \). So our expression becomes \( \frac{(1 - \cos^2 2x)^2}{\cos^4 2x} \).

Step 4: Solving the expression

Expanding \( (1 - \cos^2 2x)^2 \) and simplifying, the result is \( \frac{1 - 2\cos^2 2x + \cos^4 2x}{\cos^4 2x} \).

Key Concepts

Power-Reducing Formulas

Power-reducing formulas are a helpful tool when we work with trigonometric functions. They allow us to express powers of trigonometric functions, like sine and cosine, using first powers instead. These formulas stem from fundamental trigonometric identities.

The key power-reducing formulas are:

• For sine: \[\sin^2 x = \frac{1 - \cos 2x}{2}\]
• For cosine: \[\cos^2 x = \frac{1 + \cos 2x}{2}\]

They help us simplify expressions where we have sine or cosine raised to higher powers, by transforming them into identities. In this problem, we used the power-reducing formula for sine to change \(\sin^4 2x\) into \((1 - \cos^2 2x)^2\). This alteration is crucial because it assists in rewriting complex expressions in simpler terms.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to its sides. The most common trigonometric functions are sine, cosine, and tangent. In this exercise, our focus is on the tangent function, which can be expressed using sine and cosine functions.

The tangent of an angle \(x\) is the ratio of the sine and cosine of that angle. Thus, \(\tan x = \frac{\sin x}{\cos x}\). For our problem, we are given \(\tan^4 2x\). We start by using the identity, substituting \(\tan x\) with \(\frac{\sin x}{\cos x}\), which leads us to \(\left( \frac{\sin 2x}{\cos 2x} \right)^4\). This transformation uses the definition of tangent and breaks it down into more manageable parts by involving only sine and cosine, making it easier to apply power-reducing formulas in the next steps.

Simplifying Expressions

Simplifying expressions is about breaking down complex mathematical expressions into simpler, more easily comprehended forms. By doing so, it is easier to solve and understand these expressions.

In our exercise, after substituting \(\tan 2x\) with its sine and cosine components, we expanded and simplified using power-reducing formulas. This led us to the expression \(\frac{(1 - \cos^2 2x)^2}{\cos^4 2x}\).

Expanding \((1 - \cos^2 2x)^2\) gives \(1 - 2\cos^2 2x + \cos^4 2x\). By putting this over the common denominator \(\cos^4 2x\), we achieve a fully simplified expression. This process involves rewriting functions using identities (such as power-reducing) and performing algebraic steps to break the expression down into its simplest form so that it contains only terms of the first power of the cosine function, achieving the initial goal of the problem.