Problem 17 - Algebra 2

Question

Verify that each of the following is an identity. $$ \frac{\sin \theta}{1-\cos \theta}+\frac{1-\cos \theta}{\sin \theta}=2 \csc \theta $$

Step-by-Step Solution

Answer

The identity is verified since both sides simplify to \( 2 \csc \theta \).

Step 1: Simplify the Left Side

Consider the expression on the left side of the equation: \( \frac{\sin \theta}{1-\cos \theta} + \frac{1-\cos \theta}{\sin \theta} \). We need to combine these two fractions. The common denominator is \( \sin \theta (1 - \cos \theta) \). Combine them:\[ \frac{\sin^2 \theta + (1-\cos \theta)^2}{(1-\cos \theta) \sin \theta} \]

Step 2: Expand and Simplify the Numerator

Expand the numerator \( \sin^2 \theta + (1-\cos \theta)^2 \):\[ \sin^2 \theta + (1-2\cos \theta + \cos^2 \theta) \] Combine like terms:\[ \sin^2 \theta + 1 - 2\cos \theta + \cos^2 \theta \] Recall that \( \sin^2 \theta + \cos^2 \theta = 1 \), so the expression becomes:\[ 1 + 1 - 2\cos \theta = 2(1-\cos \theta) \]

Step 3: Substitute and Simplify

Substitute the simplified numerator back into the expression:\[ \frac{2(1-\cos \theta)}{(1-\cos \theta) \sin \theta} \]Cancel the \( 1-\cos \theta \) terms from the numerator and the denominator:\[ \frac{2}{\sin \theta} \]

Step 4: Simplify using Trigonometric Identity

Recall that \( \csc \theta = \frac{1}{\sin \theta} \). Thus, we have:\[ \frac{2}{\sin \theta} = 2 \csc \theta \] Therefore, the left side simplifies to the right side of the original identity.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions and is typically denoted by \( \sin \theta \). It measures the vertical component of an angle in the unit circle, essentially reflecting the height of the point on the outer edge of the circle. This means:

• \( \sin \theta \) corresponds to the \( y \)-coordinate of a point on the unit circle, where \( \theta \) is the angle created by the point, the origin, and the point on the circle.
• In the right triangle context, \( \sin \theta \) is the ratio of the length of the opposite side to the hypotenuse.

When you see expressions involving \( \sin \theta \), it’s often paired with its complementary identity, the cosine function.

Cosine Function

The cosine function, represented as \( \cos \theta \), works hand-in-hand with the sine function. It's often explained through the context of a right triangle or the unit circle. In mathematical terms:

• \( \cos \theta \) represents the \( x \)-coordinate of a point on the unit circle.
• For a given angle \( \theta \) in a right triangle, \( \cos \theta \) is the ratio of the length of the adjacent side to the hypotenuse.

A key identity involving both functions is the Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). This relationship keeps appearing when simplifying trigonometric expressions, like in our given exercise. It elegantly combines both functions for easy simplification of expressions.

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is the reciprocal of the sine function. It’s not as commonly used as its counterparts but plays an important role in reciprocating values:

• \( \csc \theta = \frac{1}{\sin \theta} \), meaning it measures the ratio of the hypotenuse to the opposite side in a right triangle.
• Since it’s a reciprocal, when \( \sin \theta \) is very small, \( \csc \theta \) becomes very large.

In the identity from the exercise, \( \csc \theta \) is related to the simplification of the expression \( \frac{2}{\sin \theta} \), demonstrating its reciprocal property and showing how it helps complete the identity verification.