Problem 61 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about a simplification of the expression. Verify the resulting identity algebraically. $$y=\frac{1}{\cot x+1}+\frac{1}{\tan x+1}$$

Step-by-Step Solution

Answer

The simplified expression is \(y= \frac{2}{1+\tan x}\). The graph of this function resembles a cosine curve, suggesting it's equivalent to some form of a cosine function, which is confirmed algebraically using the tangent half-angle formula. Specifically, it can be shown that this function is equivalent to \(y = 1 - 2\tan^2(x/2)\), which is a cosine function after the application of a half-angle identity.

Step 1: Simplify the given expression

First, rewrite the expression \(y=\frac{1}{\cot x+1}+\frac{1}{\tan x+1}\) using the fact that cotangent is the reciprocal of tangent. This gives us the equation \(y=\frac{1}{\frac{1}{\tan x}+1}+\frac{1}{\tan x+1}\). Then simplify this further by finding a common denominator, giving \(y=\frac{1}{1+\tan x}+\frac{1}{1+\tan x}\). which simplifies to \(y= \frac{2}{1+\tan x}\)

Step 2: Graph the function

Use a graphing utility to graph the function \(y= \frac{2}{1+\tan x}\). Observe how the curve behaves.

Step 3: Make a conjecture

Observe that the graph appears to resemble a standard cosine curve aside from its amplitude, phase and vertical shifts. Therefore, conjecture that this equation could be equivalent to some form of a cosine function.

Step 4: Verify algebraically

To verify the conjecture, we'll make use of the tangent half-angle formula which is \(\tan(x/2) = \frac{1 - \cos(x)}{\sin(x)}\). Arrange this formula for cosine to get \(\cos(x) = 1 - 2\tan^2(x/2)\). Applying this to our function \(y= \frac{2}{1+\tan x}\) after letting \(u = \tan(x/2)\) gives an equivalent cosine function.

Key Concepts

Cotangent and Tangent Identities

Trigonometric functions are often intertwined through various identities that allow for simplification and transformations. Specifically, the cotangent and tangent are reciprocals of one another. Mathematically, we express this as:
\[\begin{equation}\cot(x) = \frac{1}{\tan(x)}\end{equation}\]
And conversely,
\[\begin{equation}\tan(x) = \frac{1}{\cot(x)}\end{equation}\]
These fundamental identities are crucial for simplifying complex trigonometric expressions. For example, in the exercise, converting the cotangent to the reciprocal of tangent allowed us to combine terms and further simplify the equation.

Tangent Half-Angle Formula

The tangent half-angle formula is a valuable tool in trigonometry, especially for solving and simplifying equations involving trigonometric functions. It relates the tangent of half of an angle to the sine and cosine functions of the full angle:
\[\begin{equation}\tan\left(\frac{x}{2}\right) = \frac{1 - \cos(x)}{\sin(x)}\end{equation}\]
In the context of graph simplifications and function transformations, this formula can be rearranged to solve for cosine, as shown in step 4 of the solution provided. By employing this formula, we can convert a function involving tangent to one involving sine or cosine, allowing us to identify patterns and make conjectures about a function's graph, as seen in the exercise with the expression that closely resembled a cosine curve.

Trigonometric Function Simplification

Simplifying trigonometric functions is a process that often encompasses various steps and utilizes different properties and identities of trigonometry. The goal is to rewrite the function in a form that is easier to work with or understand.
In our problem, simplification was achieved by first utilizing reciprocals to rewrite the cotangent term as \[\begin{equation}\frac{1}{\cot x} = \tan x\end{equation}\]
Then, finding a common denominator allowed us to merge the two terms into a single fraction. This kind of simplification is essential in calculus, as it can make the difference between an intractably complex integral or derivative and a manageable one.

Verifying Trigonometric Identities

Verifying trigonometric identities involves showing that two different trigonometric expressions are equivalent, often by simplifying or transforming one or both sides of the equation. Techniques used in verification include algebraic manipulation, using known identities, and sometimes substituting trigonometric formulas.
In our example, after graphing and conjecturing the function's identity, algebraic verification was used to confirm the graphically observed similarity to a cosine function. This step is essential, because visual resemblance alone cannot conclusively prove that two functions are identical. Algebraic verification provides the mathematical rigor needed to confirm that our visual and intuitive understanding is correct.