Problem 9 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Use the power-reducing formulas for \(\sin ^{2} x\) and \(\cos ^{2} x\) to show that $$ \tan ^{2} x=\frac{1-\cos 2 x}{1+\cos 2 x} $$

Step-by-Step Solution

Answer

The power reducing identities for \(\sin ^{2} x\) and \(\cos ^{2} x\) can successfully be used to prove that \(\tan ^{2} x=\frac{1-\cos 2 x}{1+\cos 2 x}\).

Step 1: Write down the given equation

To start with, write down the equation to be proven which is \(\tan ^{2} x=\frac{1-\cos 2 x}{1+\cos 2 x}\)

Step 2: Representation of \(\tan ^{2} x\)

Represent \(\tan ^{2} x\) in terms of \(\sin ^{2} x\) and \(\cos ^{2} x\) as \(\tan ^{2} x = \frac{\sin ^{2} x}{\cos ^{2} x}\)

Step 3: Substitution using Power Reducing Formulas

Substitute the power reducing formulas: \(\sin ^{2} x = \frac{1 - \cos 2x}{2}\) and \(\cos ^{2} x = \frac{1 + \cos 2x}{2}\) into the equation. This will result in: \(\tan ^{2} x = \frac{\frac{1 - \cos 2x}{2}}{\frac{1 + \cos 2x}{2}}\)

Step 4: Simplify the Equation

Simplify the equation by multiplying the numerator and the denominator by 2. This results in: \(\tan ^{2} x=\frac{1-\cos 2 x}{1+\cos 2 x}\)

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the trigonometric functions (sine, cosine, tangent, etc.) to one another. These identities are crucial for simplifying expressions and solving trigonometric equations. Understanding trigonometric identities helps students to transform complex problems into simpler ones that are more easily solved.

Power-reducing formulas are a type of trigonometric identity that express powers of sine and cosine in terms of functions of double angles. For instance, the power-reducing formulas for \(\sin^2 x\) and \(\cos^2 x\) are:\[\sin^2 x = \frac{1 - \cos 2x}{2}\] and \[\cos^2 x = \frac{1 + \cos 2x}{2}\]. These formulas are particularly handy in solving integrals and trigonometric equations, as they reduce the power of the functions and make the algebra more manageable.

Tangent Function

The tangent function, often denoted as \(\tan x\), is one of the six fundamental trigonometric functions. It is defined as the ratio of the sine to the cosine function: \[\tan x = \frac{\sin x}{\cos x}\]. This function reflects the relationship between the angles and sides of a right triangle but also has interpretations in the unit circle.

The identity \(\tan^2 x = \frac{\sin^2 x}{\cos^2 x}\) is derived from dividing the square of the sine function by the square of the cosine function, which is valuable when solving trigonometric equations. When paired with power-reducing formulas, the tangent function helps to simplify expressions that involve squared terms, as seen in the exercise where \(\tan^2 x\) is expressed in terms of the cosine function using a double angle.

Cosine Function

The cosine function, denoted as \(\cos x\), is another primary trigonometric function, which represents the x-coordinate of a point on the unit circle at an angle \(x\) from the positive x-axis. It's commonly used to describe wave patterns, oscillations, and to determine the adjacent side of a right triangle given the hypotenuse.

In our exercise, understanding the cosine function was essential as we used its power-reducing formula to transform \(\cos^2 x\) into a function of \(\cos 2x\). The relationship between \(\cos x\) and \(\cos 2x\) was a key step in proving the given equation for \(\tan^2 x\), by substituting the power-reducing expression for \(\cos^2 x\) into the tangent identity.

Sine Function

Similar to the cosine function, the sine function, \(\sin x\), represents the y-coordinate of a point on the unit circle and is crucial in determining the opposite side of a right triangle when the hypotenuse is known. It's essential for describing periodic phenomena, like sound waves or the position of a pendulum.

In the context of the exercise, by utilizing the power-reducing formula to portray \(\sin^2 x\) in terms of \(\cos 2x\), we could express the squared tangent function in a more manageable way. The effectiveness of the exercise is in the insightful application of these formulas to reaffirm trigonometric identities such as \(\tan^2 x\), essentially reinforcing the interconnection between the sine and cosine functions.