Problem 20 - Precalculus: Functions and Graphs

Question

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: \( \frac{1+\sec 4 x}{\sin 4 x+\tan 4 x}=\csc 4 x \).

Step 1: Understanding the Given Identity

We are tasked with verifying the trigonometric identity: \( \frac{1+\sec 4x}{\sin 4x+\tan 4x}=\csc 4x \). This means that the left-hand side (LHS) should simplify to match the right-hand side (RHS).

Step 2: Rewrite Secant and Tangent

Start by expressing \( \sec 4x \) and \( \tan 4x \) in terms of sine and cosine:- \( \sec 4x = \frac{1}{\cos 4x} \)- \( \tan 4x = \frac{\sin 4x}{\cos 4x} \).

Step 3: Substitute Using Trigonometric Definitions

Substitute \( \sec 4x \) and \( \tan 4x \) into the LHS:\[ \frac{1 + \frac{1}{\cos 4x}}{\sin 4x + \frac{\sin 4x}{\cos 4x}}. \]

Step 4: Simplify the Numerator and Denominator

Simplify the numerator and denominator:- Numerator: \( 1 + \frac{1}{\cos 4x} \) becomes \( \frac{\cos 4x + 1}{\cos 4x} \).- Denominator: \( \sin 4x + \frac{\sin 4x}{\cos 4x} \) becomes \( \frac{\sin 4x \cos 4x + \sin 4x}{\cos 4x} \).

Step 5: Simplify the Fraction

Form the main fraction by combining the simplified parts:\[ \frac{\frac{\cos 4x + 1}{\cos 4x}}{\frac{\sin 4x (\cos 4x + 1)}{\cos 4x}}. \]

Step 6: Cancel Common Factors

Cancel \( \cos 4x + 1 \) from both the numerator and the denominator to simplify:\[ \frac{1}{\sin 4x} \].

Step 7: Recognize the Simplified Form

The expression \( \frac{1}{\sin 4x} \) is precisely \( \csc 4x \). Thus, LHS = RHS, verifying the identity.

Key Concepts

Secant

The secant function, denoted as \( \sec x \), is a fundamental trigonometric identity. It originates from the reciprocal relationship with the cosine function.
In other words, secant is defined as:\[ \sec x = \frac{1}{\cos x} \]Understanding secant begins with knowing that cosine represents the ratio of the adjacent side over the hypotenuse in a right-angle triangle.
Hence, secant flips this ratio, making it the hypotenuse over the adjacent side. This can be particularly useful when simplifying dead-end calculations involving cosine.

• **Key fact:** Secant is undefined where cosine is zero. This occurs at \( \frac{\pi}{2} + k\pi \) for any integer \( k \).
• **Real-world application:** Secant helps in analyzing the behavior of waves and oscillations in physics.

Grasping secant’s role in trigonometric identities, such as in the problem \( 1 + \sec 4x \), is crucial for algebraic manipulation and verifying equations.

Tangent

Tangent, represented by \( \tan x \), is a trigonometric function derived from the sine and cosine functions. It is defined by the ratio of sine to cosine as follows:\[ \tan x = \frac{\sin x}{\cos x} \]This expression is vital because it captures both sine and cosine properties. When visualizing a right triangle, tangent equates to the opposite side over the adjacent side.
Understanding tangent allows us to manipulate expressions efficiently. For example, in the problem given, expressing \( \tan 4x \) in terms of sine and cosine, simplifies the process:

• **Behavior:** Tangent can increase or decrease dramatically since it becomes undefined where cosine is zero.
• **Periodic nature:** Tangent’s function repeats every \( \pi \), allowing predictability in complex trigonometric calculations.

These aspects make tangent powerful, especially in identity verifications and solving trigonometric equations.

Cosecant

Cosecant, noted as \( \csc x \), is the reciprocal of the sine function. It is defined by:\[ \csc x = \frac{1}{\sin x} \]In practical terms, if sine represents the ratio of the opposite side over the hypotenuse in a triangle, cosecant flips this, indicating the hypotenuse over the opposite side.
This reciprocal nature is key in transforming and verifying trigonometric identities. For instance, in our exercise, the expression simplifies to \( \csc 4x \), highlighting that the left-hand side (LHS) equals the right-hand side (RHS).

• **Properties:** Cosecant is undefined where sine is zero. This happens at integer multiples of \( \pi \).
• **Applications:** Cosecant is essential in wave functions and geometrical problems involving arcs and circles.

Mastering cosecant provides you with an edge in simplifying complex trigonometric expressions and in addressing exhaustive homework challenges.