Problem 141 - Calculus Volume 1

Question

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. $$\frac{\tan ^{2} x}{\sec ^{2} x}$$

Step-by-Step Solution

Answer

The simplified expression is \( \sin^2 x \).

Step 1: Express Tangent in Sine and Cosine

The tangent function can be written in terms of sine and cosine: \( \tan x = \frac{\sin x}{\cos x} \). Therefore, \( \tan^2 x = \left( \frac{\sin x}{\cos x} \right)^2 = \frac{\sin^2 x}{\cos^2 x} \).

Step 2: Express Secant in Sine and Cosine

The secant function can be expressed with cosine: \( \sec x = \frac{1}{\cos x} \). Thus, \( \sec^2 x = \left( \frac{1}{\cos x} \right)^2 = \frac{1}{\cos^2 x} \).

Step 3: Substitute Expressions in Original Fraction

Substitute the expressions from Steps 1 and 2 into the original problem: \( \frac{\tan^2 x}{\sec^2 x} = \frac{\frac{\sin^2 x}{\cos^2 x}}{\frac{1}{\cos^2 x}} \).

Step 4: Simplify by Inverting and Multiplying

When dividing by a fraction, you can multiply by its reciprocal. Thus, \( \frac{\frac{\sin^2 x}{\cos^2 x}}{\frac{1}{\cos^2 x}} = \frac{\sin^2 x}{\cos^2 x} \cdot \frac{\cos^2 x}{1} \).

Step 5: Cancel and Simplify

Cancel \(\cos^2 x\) in the numerator and the denominator to simplify: \( \sin^2 x \cdot 1 = \sin^2 x \).

Key Concepts

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that arise from the sides of a right triangle. The sine of an angle, typically denoted as \( \sin x \), represents the ratio of the length of the side opposite the angle to the hypotenuse of the triangle. Conversely, the cosine of an angle, \( \cos x \), describes the ratio of the length of the adjacent side to the hypotenuse.

These functions form the basis for understanding other trigonometric identities, such as tangent and secant. Each function can be expressed in terms of sine and cosine, which simplifies the process of manipulating trigonometric expressions. In such expressions, you often use identities like \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \).

• Sine and cosine are periodic functions with a period of \( 2\pi \).
• Both functions range in value from -1 to 1.
• They are useful for simplifying complex trigonometric expressions.

Tangent and Secant

The tangent and secant functions are derived from sine and cosine, serving specific purposes in trigonometry. The tangent function, \( \tan x \), is calculated as the ratio of sine to cosine \( \tan x = \frac{\sin x}{\cos x} \). It's particularly useful in situations where you need to relate angles in right triangles to side length ratios other than the basic sine and cosine.

The secant function is the reciprocal of cosine, denoted as \( \sec x = \frac{1}{\cos x} \). This identity becomes especially useful in trigonometric simplification when the division by cosine needs to be avoided.

• Tangent is undefined where cosine is zero since it involves division by cosine.
• Secant is undefined where cosine equals zero, leading to infinite values.
• Understanding these functions' roles and transformations helps in solving complex trigonometric identities.

Simplification of Expressions

Trigonometric simplification is a technique used to make expressions more manageable by using identities and standard function definitions to reduce complexity. The exercise involving \( \frac{\tan^2 x}{\sec^2 x} \) showcases this simplification, where both tangent and secant are expressed in terms of sine and cosine.

By connecting these terms back to sine and cosine, you open up possibilities to cancel elements and simplify the expression efficiently. For instance, with each function rewritten in sine and cosine, you reach an expression that is easier to understand and evaluate. Here are some steps typically followed:

• Express complex trigonometric functions in terms of sine and cosine.
• Simplify fractions by multiplying by the reciprocal.
• Cancel out common terms where possible to simplify further.
• Consistently validate your simplified result against the original expression.