Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \frac{1+\sec 4 x}{\sin 4 x+\tan 4 x}=\csc 4 x $$

Step-by-Step Solution

Answer

The identity is verified as both sides equal \( \csc 4x \).

Step 1: Introduction to Identities

To verify this trigonometric identity, we will manipulate the left-hand side to show that it is equal to the right-hand side, which is \( \csc 4x \). We will use some known trigonometric identities during this process.

Step 1: Rewrite Trigonometric Functions

Start by expressing each trigonometric function in terms of sine and cosine. Recall that \( \sec 4x = \frac{1}{\cos 4x} \) and \( \tan 4x = \frac{\sin 4x}{\cos 4x} \). Substitute these into the left-hand side:\[\frac{1 + \frac{1}{\cos 4x}}{\sin 4x + \frac{\sin 4x}{\cos 4x}}\]

Step 2: Simplify the Numerator

Simplify the numerator:\[ 1 + \frac{1}{\cos 4x} = \frac{\cos 4x + 1}{\cos 4x}\]

Step 3: Simplify the Denominator

Simplify the denominator by adding the fractions:\[ \sin 4x + \frac{\sin 4x}{\cos 4x} = \frac{\sin 4x \cdot \cos 4x + \sin 4x}{\cos 4x} = \frac{\sin 4x (\cos 4x + 1)}{\cos 4x} \]

Step 4: Combine the Fractions

With the fractions prepared, combine the numerator and denominator:\[\frac{\frac{\cos 4x + 1}{\cos 4x}}{\frac{\sin 4x (\cos 4x + 1)}{\cos 4x}} \]This simplifies further to:\[ \frac{\cos 4x + 1}{\sin 4x (\cos 4x + 1)} \]

Step 5: Simplify by Canceling Common Terms

Since \( \cos 4x + 1 \) appears in both the numerator and the denominator, you can cancel these out, leading to:\[\frac{1}{\sin 4x}\]

Step 7: Final Step: Interpret the Result

The expression \( \frac{1}{\sin 4x} \) is equivalent to \( \csc 4x \), confirming that the given identity holds true.

Key Concepts

Secant and Cosecant

Secant and cosecant are two of the basic trigonometric functions, often appearing in identity problems. These functions are derived from cosine and sine, respectively. Understanding them is key to solving many trigonometric exercises.

Secant, denoted as \( \sec \theta \), is the reciprocal of the cosine function:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• It is undefined wherever \( \cos \theta = 0 \)

Cosecant, written as \( \csc \theta \), is the reciprocal of sine:

• \( \csc \theta = \frac{1}{\sin \theta} \)
• It is undefined wherever \( \sin \theta = 0 \)

This reciprocal relationship means that these functions extend the values of sine and cosine to those outside the unit circle. Understanding these basic definitions is vital when verifying trigonometric identities, as they often need to be rewritten in terms of sine and cosine to simplify and verify an equation.

Sine and Cosine

Sine and cosine are the cornerstone functions in trigonometry, essential for understanding various identities. They describe the fundamental relationship between the angles and sides of a right triangle, yet their usefulness extends well beyond triangles.

**Key Properties of Sine and Cosine:**

• Sine function, \( \sin \theta \), denotes the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.
• Cosine function, \( \cos \theta \), is the ratio of the adjacent side to the hypotenuse.
• These functions are periodic, with a period of \( 2\pi \).
• Both functions are smooth and continuous, oscillating between -1 and 1 across the circle.

Deep knowledge of these functions is necessary when converting complex trigonometric expressions into equivalent, simplified forms. For example, the secant and tangent are best understood through their definitions that utilize sine and cosine. By breaking down these more complex functions into sine and cosine, we can unravel more complex identities.

Verifying Identities

Verifying identities in trigonometry involves proving that two sides of an equation are equivalent for all values within their domains. This process often requires a strategic approach to rewriting and simplifying expressions by applying known trigonometric identities and properties.

**General Steps to Verifying Identities:**

• Start by focusing efforts on one side of the identity, usually the more complex side, and gradually transform it to look like the other.
• Utilize basic identities like \( \sin \theta = \frac{1}{\csc \theta} \) and \( \cos \theta = \frac{1}{\sec \theta} \).
• Simplify the expressions by factoring, distributing, and finding common denominators as needed.
• Cancel terms carefully, ensuring that you don't eliminate potential solutions by dismissing zero-divisors.
• Finally, reach a conclusion by showing the transformation results in an equivalent expression.

The ultimate goal is to manipulate the trigonometric expressions to confidently prove that both sides of the identity are equal, irrespective of the angle.