Problem 15 - Contemporary Precalculus

Question

Prove the identity. $$\frac{\tan x}{\sec x}=\sin x$$

Step-by-Step Solution

Answer

Question: Prove the identity \(\frac{\tan x}{\sec x}=\sin x\) Solution: To prove the given identity, follow these steps: Step 1: Rewrite the expressions in terms of sine and cosine. Replace \(\tan x\) with \(\frac{\sin x}{\cos x}\) and \(\sec x\) with \(\frac{1}{\cos x}\). Step 2: Substitute the expressions into the identity. The identity becomes: $$\frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x}}=\sin x$$ Step 3: Simplify the left side of the identity. Rewrite the expression as a product: $$\frac{\sin x}{\cos x} \cdot \frac{\cos x}{1} =\sin x$$ The \(\cos x\) in the numerator and denominator cancel out, leaving: $$\sin x=\sin x$$ Thus, the identity is proven.

Step 1: Rewrite the expressions in terms of sine and cosine.

We will rewrite the expression \(\frac{\tan x}{\sec x}\) in terms of sine and cosine. The tangent function, \(\tan x\), can be expressed as \(\frac{\sin x}{\cos x}\), while the secant function, \(\sec x\), can be expressed as \(\frac{1}{\cos x}\).

Step 2: Substitute the expressions into the identity

By substituting the expressions from step 1, our identity now becomes: $$\frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x}}=\sin x$$

Step 3: Simplify the left side of the identity

Now, we will simplify the left side of the equation. To do this, we can rewrite the expression as a product: $$\frac{\sin x}{\cos x} \cdot \frac{\cos x}{1} =\sin x$$ Notice that the \(\cos x\) in the numerator and denominator will cancel out: $$\frac{\sin x}{\cancel{\cos x}} \cdot \frac{\cancel{\cos x}}{1} =\sin x$$ Leaving us with: $$\sin x=\sin x$$ The identity has been proven.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan x \), is a fundamental trigonometric function. It is defined using the sine and cosine functions. The relationship can be expressed as:\[ \tan x = \frac{\sin x}{\cos x} \]This equation illustrates that the tangent function is the ratio of the sine of angle \( x \) to the cosine of angle \( x \). It plays a crucial role in trigonometry, helping to solve right-angled triangle problems and analyze periodic behavior.

• The tangent function is periodic with a period of \( \pi \) radians.
• It is important to note that \( \tan x \) is undefined when \( \cos x = 0 \), which occurs at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
• This function is positive in the first and third quadrants of the unit circle, and negative in the second and fourth.

The relationship \( \tan x = \frac{\sin x}{\cos x} \) is especially useful when proving identities or simplifying trigonometric expressions.

Secant Function

The secant function, denoted \( \sec x \), is another critical trigonometric function. It is the reciprocal of the cosine function. The definition is as follows:\[ \sec x = \frac{1}{\cos x} \]This implies that the secant function is undefined where the cosine function equals zero. These points occur at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.

• Since \( \sec x \) is the reciprocal of \( \cos x \), it inherits the properties related to the sign of \( \cos x \): positive in the first and fourth quadrants, negative in the second and third.
• The secant function has a period of \( 2\pi \), like the cosine function.
• Its role is vital when working with trigonometric identities, especially those requiring transformations or inversions of cosine values.

Because \( \sec x = \frac{1}{\cos x} \), it simplifies greatly into multiplication with \( \tan x \), especially useful in proving the identity given in the exercise.

Sine Function

The sine function, represented as \( \sin x \), is one of the primary trigonometric functions. It is critical in describing wave patterns, circular motion, and oscillations in physics and engineering.

• The sine function has a range of values from \(-1\) to \(1\).
• It is periodic with a period of \( 2\pi \), which means it repeats its values every \( 2\pi \) radians.
• The function is defined as the y-coordinate of the corresponding angle \( x \) on the unit circle.

The sine function is often used in identity proofs, as seen in the exercise provided. By expressing \( \frac{\tan x}{\sec x} \) in terms of \( \sin x \), we can directly simplify the equation. The sine function's basic identity helps streamline the verification of other complex identities.