Problem 24 - Precalculus with Calculus Previews

Question

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

Step-by-Step Solution

Answer

The identity is verified as both sides equal \( \frac{1+\sin x}{\cos x} \).

Step 1: Write the identity in terms of sine and cosine

We need to express \( \sec x \) and \( \tan x \) in terms of sine and cosine to simplify the right side of the equation. \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \). So, the right side becomes: \(\frac{1}{\cos x} + \frac{\sin x}{\cos x}\)

Step 2: Combine the fractions on the right side

The right side has two fractions \( \frac{1}{\cos x} + \frac{\sin x}{\cos x} \). Since they have the same denominator, they can be combined as \(\frac{1 + \sin x}{\cos x}\). Now the expression on the right side matches the left side of the identity.

Step 3: Conclusion - Verify that both sides are equal

After simplifying, both sides of the given identity are \(\frac{1 + \sin x}{\cos x}\). Since both sides of the equation are equal, the identity is verified.

Key Concepts

Sine and Cosine Functions

The sine and cosine functions are two of the most fundamental trigonometric functions and play a key role in understanding periodic phenomena, such as waves. The sine function, noted as \( \sin x \), represents the y-coordinate of a point on the unit circle as you move around it. Similarly, the cosine function, \( \cos x \), represents the x-coordinate on the unit circle. Together, they describe the position of any point based on an angle \( x \).

• **Sine Function:** Measures the vertical distance from the x-axis to the circle.
• **Cosine Function:** Measures the horizontal distance from the y-axis to the circle.

The two functions are linked through a key identity: \( \sin^2 x + \cos^2 x = 1 \). This is known as the Pythagorean identity and is essential for simplifying and solving trigonometric equations.
In the context of our problem, the identity was rewritten in terms of sine and cosine to aid in simplification.

Secant and Tangent Functions

The secant and tangent functions are closely related to the sine and cosine functions but offer a unique perspective in trigonometry. The secant function, \( \sec x \), is defined as the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). Meanwhile, the tangent function, \( \tan x \), is the ratio of the sine to the cosine function: \( \tan x = \frac{\sin x}{\cos x} \).

• **Secant Function:** Essentially "stretches" the cosine function, showing how far a point is from the circle's center horizontally.
• **Tangent Function:** Shows the "slope" or steepness of the angle, relating height to base in a triangle formed within the circle.

These functions often come into play when dealing with identities because they naturally extend sinusoidal and cosinusoidal relations. For instance, by converting secant and tangent to expressions of sine and cosine, as done in verifying our identity, you can simplify complex trigonometric equations.

Verification of Trigonometric Identities

Verification of trigonometric identities involves proving that two sides of an equation are equal by manipulating them algebraically. This process often requires converting complex trigonometric functions into terms of simpler ones, such as sine and cosine, which is a fundamental step shown in the exercise.
When verifying an identity, it's important to:

• **Convert back to basics:** Rewrite functions in their simpler forms.
• **Combine like terms when possible:** As shown, combining fractions with the same denominator can simplify the expression significantly.
• **Check for equality:** Ensure that after simplification, both sides of the equation match.

This exercise exemplifies using fundamental transformations to equate both sides of a trigonometric identity. By decomposing and then simplifying the functions into sine and cosine, each step draws us nearer to verifying the identity conclusively.