Problem 68 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\frac{1+\cos ^{2} 3 \theta}{\sin ^{2} 3 \theta}=2 \csc ^{2} 3 \theta-1$$

Step-by-Step Solution

Answer

Yes, the identity is verified.

Step 1: Understanding the Identity

The identity to verify is \(\frac{1+\cos^2 3\theta}{\sin^2 3\theta} = 2 \csc^2 3\theta - 1\). We aim to transform the left-hand side into the right-hand side.

Step 2: Using the Pythagorean Identity

Recall the Pythagorean identity: \(\sin^2 3\theta = 1 - \cos^2 3\theta\). This identity is helpful for simplifying terms involving trigonometric functions.

Step 3: Substitute and Simplify

Substitute \(\sin^2 3\theta = 1 - \cos^2 3\theta\) into the left-hand side:\[\frac{1+\cos^2 3\theta}{1-\cos^2 3\theta}.\]This expression represents a difference of squares in the denominator.

Step 4: Rewrite as Cotangent and Cosecant

The denominator simplifies to \(\sin^2 3\theta\), which is the same as \(1 - \cos^2 3\theta\). Notice the form: \[1 + \cos^2 3\theta = (1 + \cos 3\theta)(1 - \cos 3\theta) + \cos^2 3\theta = \sin^2 3\theta + 1 = \csc^2 3\theta.\]Consequently, it simplifies the expression to\[= \frac{1}{\sin^2 3\theta}(\sin^2 3\theta + 1 - 1) = \frac{\sin^2 3\theta + \cos^2 3\theta}{\sin^2 3\theta},\]where the numerator is 1, simplifying this to \(\csc^2 3\theta\).

Step 5: Final Evaluation

Now the expression becomes \(\csc^2 3\theta\). Using the identity \(\csc^2 3\theta = 1 + \cot^2 3\theta\):\[2\csc^2 3\theta - 1 = 2 \cdot \left(1 + \cot^2 3\theta\right) - 1 = 2\csc^2 3\theta - 1.\]Hence, this matches the originally provided right-hand side of the identity.

Key Concepts

Pythagorean Identity in Trigonometry

The Pythagorean identity is a fundamental concept in trigonometry. It establishes a relationship between the sine and cosine of an angle. The identity is given by the equation:\[\sin^2 \theta + \cos^2 \theta = 1\]This identity is pivotal because it allows us to express one trigonometric function in terms of another. For example, if you know the value of \(\cos^2 \theta\), you can find \(\sin^2 \theta\) using this identity:* \(\sin^2 \theta = 1 - \cos^2 \theta\)* \(\cos^2 \theta = 1 - \sin^2 \theta\)In cases where you have expressions like \(1 + \cos^2 3\theta\), as seen in this exercise, utilizing the Pythagorean identity can simplify the problem. By substituting \(\sin^2 3\theta = 1 - \cos^2 3\theta\), we transformed the expression into a manageable form. This substitution is especially useful when verifying trigonometric identities or simplifying complex trigonometric expressions.

Understanding the Cosecant Function

The cosecant function, designated as \(\csc \theta\), is the reciprocal of the sine function. It is defined as:\[\csc \theta = \frac{1}{\sin \theta}\]Cosecant is considered an important function in trigonometry, especially when dealing with identities and transformations. Since its expression involves the sine function in the denominator, \(\sin \theta\) must not be zero. This stipulation is crucial when solving equations to avoid undefined expressions.In the identity given in our exercise, we encountered the term \(\csc^2 3\theta\), which denotes the cosecant function squared. Recognizing and rewriting expressions like \(2\csc^2 3\theta - 1\) becomes significantly easier when understanding their relationship to basic trigonometric functions.By expressing the original identity in terms of \(\csc\), we could simplify and verify the transformation from the left-hand side to the right-hand side, confirming the identity successfully.

The Role of the Cotangent Function

The cotangent function, represented by \(\cot \theta\), is the reciprocal of the tangent function and closely relates to both sine and cosine:\[\cot \theta = \frac{\cos \theta}{\sin \theta}\]Like the cosecant function, \(\cot \theta\) can become undefined if the sine component in its denominator is zero. The cotangent frequently appears in identities such as \(1 + \cot^2 \theta = \csc^2 \theta\). This is particularly useful when expressing one trigonometric function in terms of others, as seen in the exercise where \(\csc^2 3\theta = 1 + \cot^2 3\theta\).Understanding these relationships allows for revealing the true nature of the identity and how each component interacts with others. Simplifying expressions with cotangent involves a clear comprehension of its foundational relationship with sine and cosine, leading to smoothing out complex identities effectively.