Problem 39 - Algebra and Trigonometry

Question

Verify the identity. $$ (1-\cos \beta)(1+\cos \beta)=\frac{1}{\csc ^{2} \beta} $$

Step-by-Step Solution

Answer

The identity is verified; both sides equal \(\sin^2 \beta\).

Step 1: Understanding the Identity

The exercise asks us to verify that \((1- ext{cos} \, \beta)(1+\text{cos} \, \beta)=\frac{1}{\csc^{2} \beta}\). To prove this identity, we need to simplify the left side and show it equals the right side.

Step 2: Simplifying the Left Side

Use the difference of squares formula \((a-b)(a+b) = a^2 - b^2\). Here, \(a = 1\) and \(b = \cos \beta\), so\((1- ext{cos} \, \beta)(1+\text{cos} \, \beta) = 1^2 - \cos^2 \beta = 1 - \cos^2 \beta\).

Step 3: Using a Pythagorean Identity

Recall the Pythagorean identity \(\sin^2 \beta + \cos^2 \beta = 1\). Rearrange it to express \(\sin^2 \beta\):\(\sin^2 \beta = 1 - \cos^2 \beta\). So, replace \(1 - \cos^2 \beta\) with \(\sin^2 \beta\).

Step 4: Expressing the Right Side

The right side of the identity is \(\frac{1}{\csc^2 \beta}\). Note that \(\csc \beta = \frac{1}{\sin \beta}\), therefore \(\csc^2 \beta = \frac{1}{\sin^2 \beta}\). Thus, \(\frac{1}{\csc^2 \beta} = \sin^2 \beta\).

Step 5: Conclusion

Both sides have been simplified to \(\sin^2 \beta\). Therefore, the original identity \((1-\cos \beta)(1+\cos \beta)=\frac{1}{\csc ^{2} \beta}\) is verified.

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic identity. It states that for any two numbers, say \(a\) and \(b\), the expression \((a-b)(a+b)\) is equal to \(a^2 - b^2\). This identity simplifies multiplication by transforming it into a subtraction of squares, which is often easier to compute or further simplify.

In the context of the exercise, we use this identity to simplify the left side of the initial equation, \((1 - \cos \beta)(1 + \cos \beta)\). Here, we identify \(a = 1\) and \(b = \cos \beta\).

• We apply the difference of squares: \((1 - \cos \beta)(1 + \cos \beta) = 1^2 - (\cos \beta)^2\).
• This results in \(1 - \cos^2 \beta\), which is a simplification useful for further steps.

This step paves the way for invoking trigonometric identities, connecting algebra with trigonometry to progress towards the solution.

Pythagorean Identity

The Pythagorean identity is one of the most important tools in trigonometry. It relates the squares of the sine and cosine functions and is given by \(\sin^2 \beta + \cos^2 \beta = 1\). This identity stems from the Pythagorean theorem, hence its name.

In the exercise, after applying the difference of squares, we arrive at the term \(1 - \cos^2 \beta\). By rearranging the Pythagorean identity, we find:

• \(\sin^2 \beta = 1 - \cos^2 \beta\).

This substitution allows us to replace \(1 - \cos^2 \beta\) with \(\sin^2 \beta\), effectively simplifying the left side to a simple trigonometric function. This transformation is crucial, as it creates a direct path to compare with the right side of the equation, ultimately leading to verification of the identity.

Cosecant Function

The cosecant function, denoted as \(\csc \beta\), is the reciprocal of the sine function. Its definition is \(\csc \beta = \frac{1}{\sin \beta}\). As a result, when we consider \(\csc^2 \beta\), it becomes \(\left(\frac{1}{\sin \beta}\right)^2 = \frac{1}{\sin^2 \beta}\).

The right side of the original identity involves \(\frac{1}{\csc^2 \beta}\). To simplify, we can rewrite:

• \(\frac{1}{\csc^2 \beta} = \sin^2 \beta\).

This simplification shows that the expression \(\frac{1}{\csc^2 \beta}\) directly equals \(\sin^2 \beta\).

Therefore, when both sides of the original equation simplify to \(\sin^2 \beta\), we confirm the identity is true. Understanding how the cosecant function operates and its relationship to the sine function is vital for seamless transitions between identities, contributing to a clearer understanding of trigonometric proofs.