Problem 9 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\frac{1}{1-\cos \gamma}+\frac{1}{1+\cos \gamma}=2 \csc ^{2} \gamma$$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \(2 \csc^2 \gamma\).

Step 1: Rewriting in Terms of Sine

First, let's rewrite the given identity using the Pythagorean identity for cosine: \(1 - \cos^2 \gamma = \sin^2 \gamma\). We'll start with the left-hand side of the equation: \(\frac{1}{1 - \cos \gamma} + \frac{1}{1 + \cos \gamma}.\)

Step 2: Simplifying the Denominator

Next, find a common denominator for the fractions on the left-hand side. Notice that \((1 - \cos \gamma)(1 + \cos \gamma) = 1 - \cos^2 \gamma = \sin^2 \gamma\). Thus, we have:\[\frac{1}{1 - \cos \gamma} + \frac{1}{1 + \cos \gamma} = \frac{1(1 + \cos \gamma) + 1(1 - \cos \gamma)}{(1 - \cos \gamma)(1 + \cos \gamma)}.\]

Step 3: Combining Terms in the Numerator

Combine the terms in the numerator:\((1 + \cos \gamma) + (1 - \cos \gamma) = 2\).Now the fraction becomes:\[\frac{2}{\sin^2 \gamma}.\]

Step 4: Converting to Cosecant

Recognize that \(\csc^2 \gamma = \frac{1}{\sin^2 \gamma}\). Therefore, we can simplify:\[\frac{2}{\sin^2 \gamma} = 2 \csc^2 \gamma.\]

Step 5: Verification

We have shown that the left-hand side simplifies to the right-hand side, thus verifying the identity:\[\frac{1}{1 - \cos \gamma} + \frac{1}{1 + \cos \gamma} = 2 \csc^2 \gamma.\]

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the fundamental identities in trigonometry. It states that for any angle \(\gamma\), the square of the sine function plus the square of the cosine function is always equal to 1. Mathematically, it is expressed as: \[sin^2 \gamma + cos^2 \gamma = 1\]
This identity is derived from the Pythagorean theorem in the context of a unit circle, where the radius is 1. Hence, the coordinates of any point on the circle will satisfy this equation.
In our exercise, we used a derivative of this identity by expressing \(1 - \cos^2 \gamma\) as \(\sin^2 \gamma\). This allows us to manipulate and simplify expressions by replacing terms that are more complex with easier-to-handle sine and cosine functions. Understanding how to leverage the Pythagorean identity is crucial in verifying and proving other trigonometric identities.

Cosecant Function

The cosecant function \(\csc \gamma\) is the reciprocal of the sine function. This means that \(\csc \gamma = \frac{1}{\sin \gamma}\). The cosecant function becomes especially useful when dealing with expressions that involve fractions or when trying to simplify trigonometric equations containing sine functions.
In our specific problem, we simplified the term \(\frac{2}{\sin^2 \gamma}\) into \(2\csc^2 \gamma\).

• The key insight here is recognizing that when you divide by a squared sine, it's equivalent to multiplying by a squared cosecant.
• Using this information allows for straightforward simplification of the equation, leading directly to the expression on the right-hand side of our original problem.

Thus, understanding the cosecant function's role helps us efficiently verify and prove trigonometric identities involving sine.

Simplifying Fractions

Simplifying fractions is an essential algebraic skill and a key step in solving equations and verifying identities in trigonometry. When a problem involves adding fractions, like \(\frac{1}{1 - \cos \gamma} + \frac{1}{1 + \cos \gamma}\), finding a common denominator simplifies the process significantly.
Here's how you do it:

• Recognize the product \((1 - \cos \gamma)(1 + \cos \gamma)\) as a difference of squares, equating to \(\sin^2 \gamma\).
• This common denominator simplifies the addition of fractions, allowing us to express the combined fraction as \(\frac{2}{\sin^2 \gamma}\).

Simplifying further, we see that this fraction transforms directly into \(2 \csc^2 \gamma\). Mastering these techniques ensures that you can tackle a wide array of similar trigonometric problems and verify identities with ease.