Problem 46 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$(1+\tan x)^{2}=\sec ^{2} x$$

Step-by-Step Solution

Answer

Answer: No, the equation $(1+\tan x)^{2}=\sec ^{2} x$ is not an identity.

Step 1: Recall the definitions of tan(x) and sec(x)

Tan(x) is defined as the ratio of sin(x) and cos(x), i.e., \(\tan x = \frac{\sin x}{\cos x}\). Sec(x) is the reciprocal of cos(x), i.e., \(\sec x = \frac{1}{\cos x}\).

Step 2: Rewrite the equation using the definitions of tan(x) and sec(x)

We have the equation given, $$(1+\tan x)^{2}=\sec ^{2} x$$Re-write this equation using the definitions of tan(x) and sec(x) from Step 1, we get $$(1+\frac{\sin x}{\cos x})^{2}=(\frac{1}{\cos x})^{2}$$

Step 3: Simplify both sides of the equation

The left side of the equation has a common denominator, so we will simplify it by adding 1 and \(\frac{\sin x}{\cos x}\) . The right side of the equation already has the same denominator, so we don't need to do anything to it. $$\Longrightarrow \frac{(1 + \sin x)}{\cos x^{2}} = \frac{1}{\cos x^{2}}$$

Step 4: Cross multiply and simplify further equation

Cross-multiplying both sides by \(\cos x^{2}\), we get $$(1 + \sin x) = 1$$ which simplifies to $$\sin x = 0$$

Step 5: Conclude whether the given equation is an identity or not

Since the equality \(\sin x = 0\) is not true for every value of x, the given equation $$(1+\tan x)^{2}=\sec ^{2} x$$ is not an identity.

Key Concepts

Tangent Function

The tangent function is one of the six fundamental trigonometric functions. It has a special relationship with the sine and cosine functions.This relationship can be expressed as:

• The tangent function, \( an x\), is defined as the ratio of the sine function to the cosine function: \( an x = \frac{\sin x}{\cos x}\).
• The tangent function is periodic, with a period of \(\pi\), which means it repeats its values every \(\pi\) units.
• Tangent can be undefined when the cosine value (the denominator) is zero. This happens at odd multiples of \(\frac{\pi}{2}\).

The behavior of the tangent function is integral in understanding trigonometric equations and identities. It often comes into play when simplifying expressions or solving equations that involve angles.

Secant Function

The secant function, while less commonly used than sine or cosine, is important in trigonometry.

• It is defined as the reciprocal of the cosine function: \(\sec x = \frac{1}{\cos x}\).
• Secant is periodic with a period of \(2\pi\), which is the same as the cosine function.
• Just like tangent, the secant function can be undefined, specifically at the angle values where the cosine is zero.

Graphically, the secant function has vertical asymptotes wherever cosine is zero, making these crucial points to consider when dealing with secant in equations or identities.It also helps to visualize secant as a measure of how far the hypotenuse of a right triangle extends from the origin at any angle.

Trigonometric Equations

Trigonometric equations may initially look complex, but they often follow specific rules that simplify their solutions substantially. The goal is to find angle values satisfying the given equation.

• They require knowledge of trigonometric identities, like \(\sin^2 x + \cos^2 x = 1\) or \(1 + \tan^2 x = \sec^2 x\), which can help in transforming and simplifying equations.
• A typical approach could involve manipulating both sides into a common form, often involving factoring, expanding, or using reciprocal identities.
• These equations typically have solutions over a set range; because trigonometric functions are periodic, solutions can repeat over their defined periods.

It is essential to verify the results by substituting them back into the original equation.This ensures they meet the conditions set forth by the equation itself, and crucially, whether they satisfy all values of the variable involved to qualify as an identity.