Problem 61 - Precalculus with Limits

Question

In Exercises 61-64, verify the identity. \( \tan^5 x = \tan^3 x \sec^2 x - \tan^3 x \)

Step-by-Step Solution

Answer

Step by step simplification of the right side of the given identity leads to the left side, which confirms the identity \( \tan^5 x = \tan^3 x \sec^2 x - \tan^3 x \).

Step 1: Rewrite in terms of sine and cosine

The first step is to write all the functions in terms of sine and cosine. Here, \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \).

Step 2: Simplify the expression using basic algebra

Next, simplify the right side of the equation by using algebraic methods: \( \tan^3 x \sec^2 x - \tan^3 x = \tan^3 x (\sec^2 x - 1) = \tan^3 x (\tan^2 x) \). This is due to \( \sec^2 x - 1 = \tan^2 x \) from the Pythagorean identities.

Step 3: Rewrite in terms of original function

Lastly, simplify the obtained expression \( \tan^3 x \tan^2 x = \tan^5 x \) back to the original function in the identity.

Key Concepts

Pythagorean Identities

Pythagorean identities are a cornerstone in trigonometry, deeply rooted in the properties of the right triangle. They relate the squares of the sine, cosine, and tangent functions. One fundamental identity is:

• \( \sin^2 x + \cos^2 x = 1 \)
• Another identity derived from this is: \( \sec^2 x = 1 + \tan^2 x \)

The latter identity is crucial for the equation you're working with because it helps transform \( \sec^2 x - 1 \) into \( \tan^2 x \). This simplification is vital in rewriting and solving trigonometric equations efficiently.

Tangent Function

The tangent function is critical in trigonometry, often defined as the ratio of sine to cosine:

• \( \tan x = \frac{\sin x}{\cos x} \)

This relationship means when the angle approaches 90 degrees, the cosine approaches zero, causing the tangent to approach infinity. In solving trigonometric identities, rewriting tangent in terms of sine and cosine simplifies more complex equations, like those involving powers of tangent.
For instance, understanding that \( \tan^3 x \) represents \( \left( \frac{\sin x}{\cos x} \right)^3 \) allows for easier manipulation and resolution of the identity verification problem in the exercise.

Secant Function

The secant function is another trigonometric ratio, defined as the reciprocal of cosine:

• \( \sec x = \frac{1}{\cos x} \)

By transforming expressions involving secant into terms of cosine, complex trigonometric problems can become easier to handle. For instance, the given identity
\( \tan^3 x \sec^2 x - \tan^3 x \) becomes more manageable by expressing secant as \( \sec^2 x = 1 + \tan^2 x \) from the Pythagorean identities. This conversion simplifies the expressions, aiding in verifying the identity through algebraic manipulation.

Sine and Cosine Transformations

Transforming trigonometric identities with sine and cosine is an efficient strategy to solve complex problems. The sine and cosine functions form the basis from which other trigonometric functions, such as tangent and secant, are derived.
By expressing complicated functions in terms of sine and cosine, like in the identity \( \tan x = \frac{\sin x}{\cos x} \), you simplify the process of algebraic manipulation.
These transformations not only aid simplifying expressions but also serve as a verification tool by ensuring that the expressions hold true across the same trigonometric circle. They assure that you can consistently break down and understand the relationships and identities in any trigonometric equation.