Problem 92 - Algebra and Trigonometry Real Mathematics, Real People

Question

Explain why the equation is not an identity and find one value of the variable for which the equation is not true. $$1+\tan x=\sec x$$

Step-by-Step Solution

Answer

The equation is not an identity because there exists at least one value for the variable, specifically \(x = \pi/3\), for which the equation is not true.

Step 1: Define the functions

Start by getting familiar with the trigonometric functions in the equation. The tangent function is defined as \(\tan x=\frac{\sin x}{\cos x}\) and the secant function is the reciprocal of the cosine function, i.e., \(\sec x=\frac{1}{\cos x}\)

Step 2: Formulate the equation in terms of sine and cosine

Write the equation \(1 + \tan x = \sec x\) in terms of \(\sin x\) and \(\cos x\). This way, it becomes easier to solve. The equation now reads \(1 + \frac{\sin x}{\cos x} = \frac{1}{\cos x}\)

Step 3: Simplify the equation

Now find a common denominator on the left side of the equation to simplify. This implies \(\frac{\cos x + \sin x}{\cos x} = \frac{1}{\cos x}\). After simplification, the equation reads \(\cos x + \sin x = 1\)

Step 4: Find a value of x that violates the equation

Now that the equation is simplified, it's possible to find a value of x that will satisfy \(\cos x + \sin x = 1\). The trick is to find values for \(\sin x\) and \(\cos x\) such that their sum is not 1. For instance, when \(x = 60^{\circ} = \pi/3\) rad, \(\cos x = 1/2\), \(\sin x = \sqrt{3}/2\), hence the left side of the equation is \(1/2 + \sqrt{3}/2 \neq 1\). Therefore, \(x = \pi/3\) is a value for which the original equation is not true

Key Concepts

Tangent Function

The tangent function is one of the basic trigonometric functions, and it's defined as the ratio of the sine and cosine of an angle:

• \(\tan x = \frac{\sin x}{\cos x}\)

This means that to find the value of \(\tan x\) for any angle \(x\), you simply divide the sine of \(x\) by the cosine of \(x\).
This is particularly useful in solving trigonometric equations, especially when you need to transform expressions into simpler forms that involve only sine and cosine. The property of tangent being undefined when \(\cos x = 0\) is also critical, as it can affect the validity of equations like \(1 + \tan x = \sec x\). Ensure to be cautious around angles like \(90^\circ\), where \(\tan x\) doesn’t hold a finite value.

Secant Function

The secant function is less commonly discussed than sine and cosine, but it's equally important. Secant is the reciprocal of the cosine function.

• \(\sec x = \frac{1}{\cos x}\)

This relationship implies that the secant function is undefined whenever \(\cos x = 0\), similar to how tangent becomes undefined.
This property is vital because any trigonometric equation involving the secant function depends on the values where the cosine equals zero. The original equation \(1 + \tan x = \sec x\) leverages both functions, inviting the need to reassess the equation's truth using their properties.
For example, you can convert the equation to expressions involving cosine, helping you find instances where the equation is false.

Sine and Cosine Simplification

Simplifying trigonometric equations using sine and cosine often makes them easier to solve. This process involves rewriting terms like \(\tan x\) and \(\sec x\) into their sine and cosine components. The given equation \(1 + \tan x = \sec x\) can be reworked into:

• \(1 + \frac{\sin x}{\cos x} = \frac{1}{\cos x}\)

Now, by finding a common denominator for the fractions on the left, the equation simplifies further:

• \(\frac{\cos x + \sin x}{\cos x} = \frac{1}{\cos x}\)

This step is crucial for testing values that meet or violate this equation's terms. It reduces complexity, helping you verify the equation's conditions at specific angles.

Trigonometric Equation Solving

Solving trigonometric equations can often entail identifying values of the variable where the assumptions don't hold.
In the case of \(1 + \tan x = \sec x\), transforming the equation using sine and cosine functions allowed determination of such values.
Once you've simplified to an equation like \(\cos x + \sin x = 1\), it becomes easier to analyze where this condition might fail.
Consider when \(x = 60^\circ\), and check if \(\cos x + \sin x\) equals 1:

• \(\cos 60^\circ = \frac{1}{2}\) and \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)

Adding them yields \(\frac{1}{2} + \frac{\sqrt{3}}{2}\), which visibly does not equate to 1.
This analysis shows that at \(x = 60^\circ\), the original equation doesn’t hold true, showcasing the solution method for investigation in similar cases.