Problem 68 - Algebra 2

Question

Show that \(\frac{\cos x}{1-\sin ^{2} x}=\sec x\) is an identity.

Step-by-Step Solution

Answer

By using the basic properties of trigonometric functions and the Pythagorean identity, we have shown that \(\frac{\cos x}{1-\sin^{2}x}=\sec x\) is indeed an identity.

Step 1: Understand the basics

Let's begin by understanding the basic properties of trigonometric functions. Specifically, remember that \(\sec x = \frac{1}{\cos x}\) and \(\sin^2x + \cos^2x = 1\).

Step 1: Substitute for \(sec x\)

We start by replacing secant on the right-hand side by its equivalent in terms of cosine, so \(\sec x = \frac{1}{\cos x}\). So now the equation we want to prove becomes \(\frac{\cos x}{1-\sin^{2}x}=\frac{1}{\cos x}\).

Step 2: Cross-multiply

You then cross-multiply each side by \(\cos x\) and \(1-\sin^{2}x\). This simplifies the equation to \(\cos^{2}x = 1 - \sin^{2}x\).

Step 3: Use the Pythagorean identity

Next, substitute the Pythagorean identity for the right side of the equation. Remember that \(\sin^{2}x + \cos^{2}x = 1\), which rearranges to \(\cos^{2}x = 1 - \sin^{2}x\). So substituting, we get \(\cos^{2}x = \cos^{2}x\).

Key Concepts

Cosine Function

The cosine function, often denoted as \( \cos x \), is a fundamental part of trigonometry. It expresses the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Unlike the sine function, which deals with the opposite side, cosine helps us understand and analyze angles based on the adjacent side.

• Cosine values range from -1 to 1.
• The cosine function is periodic, with a period of \(2\pi\).
• It is an even function, meaning \(\cos(-x) = \cos(x)\).

Understanding the cosine function is crucial in solving trigonometric identities and equations. In our exercise, the cosine function appears in both the numerator and denominator, and understanding how it behaves is key to manipulating the equation.

Secant Function

The secant function, represented as \(\sec x\), is the reciprocal of the cosine function. This means \(\sec x = \frac{1}{\cos x}\).

• Since it is a reciprocal, secant is undefined where cosine is zero because division by zero is undefined.
• Secant values can range from 1 and above or -1 and below, reflecting the reciprocal nature of cosine's values.
• It is particularly useful in calculus and physics, where division by cosine frequently occurs.

In proving the identity \(\frac{\cos x}{1 - \sin^2 x} = \sec x\), converting secant to its cosine equivalent helps simplify the equation. Recognizing this relationship is fundamental when working with trigonometric identities.

Pythagorean Identity

The Pythagorean identity is one of the most useful identities in trigonometry, usually represented as \(\sin^2 x + \cos^2 x = 1\). This identity is derived from the equation of a unit circle, where any point \((\cos x, \sin x)\) satisfies the equation \(x^2 + y^2 = 1\).

• The identity shows the inherent relationship between sine and cosine for any angle \(x\).
• You can rearrange it to find expressions for \(\cos^2 x\) or \(\sin^2 x\), depending on what is needed.
• In our exercise, it is rewritten as \(\cos^2 x = 1 - \sin^2 x\), simplifying the original equation to a true statement.

Using the Pythagorean identity allowed us to confirm \(\cos x = 1 - \sin^2 x\), showing how essential it is in proving that trigonometric identities hold true.