Problem 71 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\tan ^{2} u-\sin ^{2} u=\tan ^{2} u \sin ^{2} u$$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \( \frac{\sin^4 u}{\cos^2 u} \).

Step 1: Express Both Sides Using Sine and Cosine

Start by writing all trigonometric functions, including the left-hand side (LHS) and the right-hand side (RHS), in terms of sine and cosine. Note: \( \tan u = \frac{\sin u}{\cos u} \).For the LHS:\[ \tan^2 u - \sin^2 u = \left(\frac{\sin^2 u}{\cos^2 u}\right) - \sin^2 u \]For the RHS:\[ \tan^2 u \sin^2 u = \left(\frac{\sin^2 u}{\cos^2 u}\right) \sin^2 u = \frac{\sin^4 u}{\cos^2 u} \]

Step 2: Simplify the Left-Hand Side

For LHS:Multiply both terms over a common denominator of \( \cos^2 u \):\[ \frac{\sin^2 u}{\cos^2 u} - \sin^2 u = \frac{\sin^2 u}{\cos^2 u} - \frac{\sin^2 u \cdot \cos^2 u}{\cos^2 u} \]Simplify the expression:\[ = \frac{\sin^2 u - \sin^2 u \cos^2 u}{\cos^2 u} = \frac{\sin^2 u (1 - \cos^2 u)}{\cos^2 u} \]

Step 3: Recognize and Use Trigonometric Identity

Recall the Pythagorean identity: \( \sin^2 u + \cos^2 u = 1 \). From this, \( 1 - \cos^2 u = \sin^2 u \).Replace \( 1 - \cos^2 u \) in the expression:\[ \frac{\sin^2 u (\sin^2 u)}{\cos^2 u} = \frac{\sin^4 u}{\cos^2 u} \]

Step 4: Compare Simplified Left-Hand Side and Right-Hand Side

Observe that the LHS simplifies to:\[ \frac{\sin^4 u}{\cos^2 u} \]The RHS is already given as:\[ \frac{\sin^4 u}{\cos^2 u} \]Since both expressions are the same, the identity is verified.

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the most fundamental relationships in trigonometry. It states that for any angle \( u \), the sum of the squares of sine and cosine equals 1, which can be written as:

• \( \sin^2 u + \cos^2 u = 1 \)

This identity is derived from the Pythagorean Theorem applied to the unit circle, where the radius is 1. Hence, it ties together the geometric interpretation of trigonometric functions on a circle.
In the given exercise, this identity is particularly useful to simplify expressions. When we encounter terms like \( 1 - \cos^2 u \), we can confidently replace them with \( \sin^2 u \) thanks to the Pythagorean identity. This is a handy tool that simplifies complex trigonometric expressions easily and efficiently.

Simplifying trigonometric expressions

Simplifying trigonometric expressions involves reducing them to a more manageable form. This often includes expressing trigonometric functions using a common denominator or substituting equivalent expressions using identities.
For example, in the original exercise, we transformed both sides of the identity using sine and cosine to find a common basis for comparison. By expressing both \( \tan u \) as \( \frac{\sin u}{\cos u} \) and \( \tan^2 u \) as \( \frac{\sin^2 u}{\cos^2 u} \), we created an opportunity to simplify terms directly.
Arranging under a common denominator and using algebraic manipulations allow students to see more clearly how the expressions relate to each other, facilitating further simplification and verification.

Sine and cosine definitions

The sine and cosine functions are the core of trigonometry. Their definitions on the unit circle are straightforward:

• \( \sin u \) is the y-coordinate of the point on the unit circle.
• \( \cos u \) is the x-coordinate of the point on the unit circle.

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane.
This means that any point \((\cos u, \sin u)\) on the unit circle directly correlates to the angle \( u \). These definitions are essential, especially when simplifying and verifying trigonometric identities, as they allow us to replace trigonometric functions with their algebraic equivalents like we did in the exercise.

Verifying identities

Verifying trigonometric identities is a process where you show that both sides of an equation are equivalent for all values of the variables involved. It involves transforming one or both sides of an equation using known identities and simplifications until they match exactly.
In the provided exercise, we verified the identity by working through each side of the equation separately, simplifying them, and finally using the Pythagorean identity to reach a common form.

• Start by expressing all trigonometric functions using sine and cosine.
• Use algebraic manipulations such as factoring, expanding, or simplifying fractions.
• Apply relevant trigonometric identities, such as the Pythagorean identity, where applicable.

Once the transformed sides are equivalent, the identity is verified. It’s similar to solving a puzzle, finding the right pieces and arranging them in the correct order to match both sides.