Problem 139 - Calculus Volume 1

Question

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. \(\tan ^{2} x+\sin x \csc x\)

Step-by-Step Solution

Answer

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

Step 1: Identify and Rewrite in Terms of Sines and Cosines

Begin by identifying each trigonometric function in the expression. We have \( \tan^2 x \) and \( \sin x \csc x \). Rewrite these functions using sines and cosines:\[ \tan x = \frac{\sin x}{\cos x} \Rightarrow \tan^2 x = \left(\frac{\sin x}{\cos x}\right)^2 = \frac{\sin^2 x}{\cos^2 x} \] and \[ \csc x = \frac{1}{\sin x} \Rightarrow \sin x \csc x = \sin x \cdot \frac{1}{\sin x} = 1 \].

Step 2: Simplify Each Part and Combine

Now that both parts of the expression are rewritten, substitute them back into the expression:\[ \tan^2 x + \sin x \csc x = \frac{\sin^2 x}{\cos^2 x} + 1 \].Notice that adding 1 to \( \frac{\sin^2 x}{\cos^2 x} \) gives you a trigonometric identity form.

Step 3: Use a Trigonometric Identity to Simplify Further

Recall the Pythagorean identity: \( 1 + \tan^2 x = \sec^2 x \). Our rewritten expression \( \frac{\sin^2 x}{\cos^2 x} + 1 \) can be seen as:\[ 1 + \frac{\sin^2 x}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x \].Thus, the expression simplifies to \( \sec^2 x \).

Key Concepts

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that express the coordinate values of a point on the unit circle. Let's break these functions down a bit:

• The sine function, written as \( \sin x \), gives the y-coordinate of a point on the unit circle.
• The cosine function, written as \( \cos x \), gives the x-coordinate of a point on the unit circle.

These functions are crucial because they allow us to express other trigonometric functions, like tangent and secant, using them. For instance, tangent can be written in terms of sine and cosine as \( \tan x = \frac{\sin x}{\cos x} \).

Rewriting expressions in terms of sine and cosine often simplifies the process because it reduces the variety of functions that you're dealing with and makes it easier to apply identities, like the Pythagorean identities. Substituting other trigonometric functions with sine and cosine can lead to simpler expressions that might elucidate hidden identities in trigonometric equations.

Tangent and Secant Functions

Tangent and secant are powerful functions closely related to sine and cosine. Understanding these can simplify complex trigonometric expressions. Let's explore further:

• The tangent function, \( \tan x \), is the ratio of sine to cosine: \( \tan x = \frac{\sin x}{\cos x} \).
• The secant function, \( \sec x \), is the reciprocal of cosine: \( \sec x = \frac{1}{\cos x} \).

In many trigonometric problems, especially when simplifying expressions like \( \tan^2 x + \sin x \csc x \), expressing the tangent in terms of sine and cosine may reveal identities. Replacing \( \tan^2 x \) with \( \frac{\sin^2 x}{\cos^2 x} \) unveils a common form tied to the Pythagorean identity.

Additionally, knowing that \( \tan^2 x + 1 = \sec^2 x \) allows combining different parts of an expression under a single identity, resulting in a much simplifed outcome like \( \sec^2 x \). This understanding can immensely help in problems involving trigonometric identities.

Pythagorean Identities

Pythagorean identities are among the most important tools in trigonometry for simplifying expressions and solving equations. They stem from the Pythagorean theorem and are applied in various forms. Here are the main identities to know:

• \( \sin^2 x + \cos^2 x = 1 \) - This relates the squares of sine and cosine to the value 1.
• \( 1 + \tan^2 x = \sec^2 x \) - A significant identity for expressions involving\( \tan x \) and \( \sec x \).
• \( 1 + \cot^2 x = \csc^2 x \) - Relates cotangent and cosecant.

In the exercise solution, the identity \( 1 + \tan^2 x = \sec^2 x \) is particularly valuable. It allows you to transform an otherwise complex expression into something quite manageable, as seen with \( \frac{\sin^2 x}{\cos^2 x} + 1 \), simplifying to \( \sec^2 x \).

Understanding these identities is key in trigonometry. They help perform complex manipulations with trigonometric expressions effortlessly, often resulting in expressions that highlight the intrinsic relationships between the trigonometric functions.