Problem 74 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{1+\tan x}{1-\tan x}=\frac{\cos x+\sin x}{\cos x-\sin x} $$

Step-by-Step Solution

Answer

The identity is verified; both sides are equal.

Step 1: Write the Given Identity

We need to verify the identity \( \frac{1+\tan x}{1-\tan x} = \frac{\cos x+\sin x}{\cos x-\sin x} \).

Step 2: Express Tan in Terms of Sine and Cosine

Recall the trigonometric identity \( \tan x = \frac{\sin x}{\cos x} \). Substitute \( \tan x \) in the left side of the identity: \( \frac{1+\frac{\sin x}{\cos x}}{1-\frac{\sin x}{\cos x}} \).

Step 3: Simplify the Left Side

Combine terms in the numerator and the denominator:Numerator: \( 1 + \frac{\sin x}{\cos x} = \frac{\cos x + \sin x}{\cos x} \).Denominator: \( 1 - \frac{\sin x}{\cos x} = \frac{\cos x - \sin x}{\cos x} \).

Step 4: Divide by the Common Denominator

The expression becomes: \( \frac{\cos x + \sin x}{\cos x} \div \frac{\cos x - \sin x}{\cos x} \). This simplifies to \( \frac{\cos x + \sin x}{\cos x - \sin x} \).

Step 5: Compare Both Sides

Now that both sides of the identity are \( \frac{\cos x + \sin x}{\cos x - \sin x} \), the identity is verified.

Key Concepts

Sine and Cosine

Sine (\(\sin\)) and cosine (\(\cos\)) are fundamental trigonometric functions that help us understand the relationships in right triangles. They also serve as building blocks for most other trigonometric concepts. In a right triangle:

• The sine function (\(\sin\)) is defined as the ratio of the length of the opposite side to the hypotenuse.
• The cosine function (\(\cos\)) is the ratio of the length of the adjacent side to the hypotenuse.

The unit circle, which is a circle with radius 1 centered at the origin, gives us another perspective on these functions.
Here, \(\sin\) and \(\cos\) correspond to the y and x coordinates of any point on the circle, respectively.
Both values range between -1 and 1 and help us describe angles beyond 0° to 90°. This way, sine and cosine functions are not limited to right triangles but offer a complete picture of periodic behavior.
Understanding these functions is crucial, as they are extensively used in verifying identities and solving trigonometric equations.

Tan Function

The tangent function, denoted by \(\tan x\), is another key trigonometric function. It is defined as the ratio of sine to cosine:\[tan x = \frac{\sin x}{\cos x}\]This property of tangent makes it very useful for solving equations involving trigonometric functions.
Since \(\tan x\) relates directly to sine and cosine, expressing tangent in terms of sine and cosine is a common step when working with identities.
For example, in verifying the identity given in the exercise, rewriting \(\tan x\) as \(\frac{\sin x}{\cos x}\) allowed us to simplify and compare both sides of the equation.
Understanding \(\tan x\) as a function that measures the steepness of a line (in terms of the unit circle) helps us incorporate it into broader applications of trigonometry, including slopes of angles and periodic functions.

Verifying Identities

Verifying identities is a crucial aspect of trigonometry, allowing us to prove that two different-looking expressions are equivalent. This process is foundational in simplifying complex trigonometric expressions and solving equations.
In the process of verifying an identity, the goal is to transform one side of the equation to make it look like the other side.
Here are some steps commonly followed in the verification process:

• Rewrite Functions: Express functions like \(\tan x\) in terms of \(\sin x\) and \(\cos x\).
• Simplify: Use algebraic techniques like combining fractions or expanding and factoring terms.
• Transform: Utilize known identities such as Pythagorean identities, angle sum and difference identities, etc.

These steps were exemplified in the solution to the original exercise, where both sides of the equation were manipulated to validate they were equal.
Practice and familiarization with a variety of identities enhance one’s ability to see connections and identify suitable approaches for different trigonometric problems.