Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity. $$ \csc 2 u=\frac{1}{2} \csc u \sec u $$

Step-by-Step Solution

Answer

The identity is verified: \( \csc 2u = \frac{1}{2\sin u \cos u} \).

Step 1: Recall the Identity for \( \csc \)

The cosecant function is the reciprocal of the sine function. Thus, \( \csc \theta = \frac{1}{\sin \theta} \). We'll use this identity to express \( \csc 2u \) in terms of sine.

Step 2: Use the Double Angle Formula for \( \sin \)

The sine double angle identity states that \( \sin 2u = 2\sin u \cos u \). Thus, \( \csc 2u = \frac{1}{\sin 2u} = \frac{1}{2\sin u \cos u} \).

Step 3: Simplify the Right Side Expression

We need to simplify \( \frac{1}{2} \csc u \sec u \) using the identities for \( \csc \) and \( \sec \). Substitute \( \csc u = \frac{1}{\sin u} \) and \( \sec u = \frac{1}{\cos u} \). Thus, \( \frac{1}{2} \csc u \sec u = \frac{1}{2} \cdot \frac{1}{\sin u} \cdot \frac{1}{\cos u} = \frac{1}{2\sin u \cos u} \).

Step 4: Verify the Equality

Both sides of the original identity simplify to the same expression \( \frac{1}{2\sin u \cos u} \). This confirms that \( \csc 2u = \frac{1}{2} \csc u \sec u \), verifying the identity.

Key Concepts

Cosecant Function

The cosecant function, often abbreviated as \( \csc \), is deeply rooted in basic trigonometry. It is one of the six fundamental trigonometric functions used in various mathematical applications. The cosecant of an angle \( \theta \) is defined as the reciprocal of the sine function. Hence, the formula is \( \csc \theta = \frac{1}{\sin \theta} \).
Understanding the relationship between the sine and cosecant functions is crucial for solving many trigonometric problems.
Since \( \sin \theta \) measures the ratio of the opposite side to the hypotenuse in a right triangle, \( \csc \theta \) measures the hypotenuse to the opposite side.
This function is not defined for angles where sine is zero, which occur at integer multiples of \( \pi \). Learning to use the cosecant function helps in simplifying and solving equations where sine functions appear in the denominator.

Sine Double Angle Formula

The sine double angle formula is an essential identity in trigonometry. It assists in simplifying expressions and solving trigonometric equations. The identity states that \( \sin 2u = 2\sin u \cos u \).
This means that the sine of twice an angle \( u \) can be determined by doubling the product of the sine and cosine of the angle \( u \).

• It can be used to express trigonometric functions of multiple angles in simpler terms.
• This identity is particularly helpful for converting products of trigonometric functions into sums, thereby making integration easier.

By rearranging and using the sine double angle formula, complex identities like those involving \( \csc 2u \) can be verified or simplified.
Knowing this formula aids in understanding deeper connections between the various trigonometric identities.

Reciprocal Identities

Reciprocal identities are a set of equations in trigonometry that express one trigonometric function as the reciprocal of another. These identities include:

• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

These identities stem from the definition of the basic trigonometric functions and provide a way to rewrite and simplify trigonometric expressions.
For instance, in verifying the identity \( \csc 2u = \frac{1}{2} \csc u \sec u \), knowing that \( \csc u = \frac{1}{\sin u} \) and \( \sec u = \frac{1}{\cos u} \) is pivotal.
Reciprocal identities allow for conversion between different trigonometric functions, which is useful when both simplifying expressions and proving other identities.
Grasping these identities is essential for solving complex equations in trigonometry.