Problem 71 - Algebra and Trigonometry

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: \( \tan^2 u - \sin^2 u = \tan^2 u \sin^2 u \).

Step 1: Express Trigonometric Functions in Terms of Sine and Cosine

We start by rewriting the tangent function in terms of sine and cosine. The identity \( \tan u = \frac{\sin u}{\cos u} \) implies that \( \tan^2 u = \left(\frac{\sin u}{\cos u}\right)^2 = \frac{\sin^2 u}{\cos^2 u} \).

Step 2: Substitute and Simplify the Left-Hand Side

Substitute \( \tan^2 u \) with \( \frac{\sin^2 u}{\cos^2 u} \) in the left-hand side (LHS) of the original equation:\[ \tan^2 u - \sin^2 u = \frac{\sin^2 u}{\cos^2 u} - \sin^2 u \]Simplify by writing \( \sin^2 u \) as \( \frac{\sin^2 u \cdot \cos^2 u}{\cos^2 u} \):\[ \frac{\sin^2 u}{\cos^2 u} - \frac{\sin^2 u \cdot \cos^2 u}{\cos^2 u} = \frac{\sin^2 u - \sin^2 u \cdot \cos^2 u}{\cos^2 u} \].

Step 3: Factor the Common Term

Factor \( \sin^2 u \) out from the numerator:\[ \frac{\sin^2 u(1 - \cos^2 u)}{\cos^2 u} \]Using the Pythagorean identity, we know that \( 1 - \cos^2 u = \sin^2 u \). Substituting this gives:\[ \frac{\sin^2 u \cdot \sin^2 u}{\cos^2 u} \].

Step 4: Simplify to Match Right-Hand Side

Combine the terms to obtain:\[ \frac{\sin^4 u}{\cos^2 u} \]Since \( \tan^2 u = \frac{\sin^2 u}{\cos^2 u} \), the expression simplifies to \( \tan^2 u \cdot \sin^2 u \), which matches the right-hand side (RHS) of the original equation.

Key Concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions, often denoted as \( \tan \). It is defined in terms of the sine and cosine functions. Specifically, the tangent of an angle \( u \) is the ratio of the sine of the angle to the cosine of the angle: \( \tan u = \frac{\sin u}{\cos u} \). This definition is derived from the geometric properties of right triangles and the unit circle.

When dealing with trigonometric identities, expressing the tangent function in terms of sine and cosine can greatly simplify the process. This is particularly useful because many identities involve products or quotients of sine and cosine, allowing us to easily see connections between different functions.

For instance, squaring the tangent function gives us \( \tan^2 u = \left(\frac{\sin u}{\cos u}\right)^2 = \frac{\sin^2 u}{\cos^2 u} \). This expression reveals how the tangent function interacts with other trigonometric terms and is essential for solving identities.

Sine and Cosine Functions

Sine and cosine are the foundational building blocks of trigonometry. These functions are crucial in understanding various properties of angles and triangles. The sine function, denoted as \( \sin \), represents the ratio of the opposite side to the hypotenuse in a right triangle. Conversely, the cosine function, \( \cos \), represents the ratio of the adjacent side to the hypotenuse.

The interdependence of sine and cosine is paramount in trigonometry due to their complementary nature. In particular, the identity \( 1 - \cos^2 u = \sin^2 u \), which is derived from the Pythagorean identity, is frequently used to simplify expressions. By recognizing how these functions relate to each other, one can effortlessly substitute and rearrange terms to achieve simplified forms of trigonometric expressions.

In many trigonometric identities, rewriting sine and cosine expressions can reveal underlying relationships, assist in factorization, and aid in simplifying complex equations.

Pythagorean Identity

The Pythagorean identity is a pivotal concept in trigonometry, showcasing the deep connection between sine and cosine functions. This identity is expressed as \( \sin^2 u + \cos^2 u = 1 \). It arises from the fundamental property of right triangles where the sum of the squares of the two shorter sides equals the square of the hypotenuse.

This identity is incredibly versatile. It allows for the transformation of trigonometric expressions and simplification of complex identities. By manipulating this identity, you can express one function in terms of the other. For example, rearranging gives \( \sin^2 u = 1 - \cos^2 u \) and \( \cos^2 u = 1 - \sin^2 u \).

These transformations help in solving trigonometric identities by providing an alternative expression depending on the requirements of the problem. In our exercise, using the Pythagorean identity to substitute \( 1 - \cos^2 u \) with \( \sin^2 u \) is a key step in verifying the given identity.