Problem 134 - Precalculus with Limits

Question

In Exercises 133 - 138, determine whether or not the equation is an identity, and give a reason for your answer. \( \cot \theta = \sqrt{\csc^2 \theta + 1} \)

Step-by-Step Solution

Answer

The given equation is not an identity because when we simplify it, the left side and right side of the expression are not equal.

Step 1: Recall the trigonometric identities

Recall that \(\cot \theta = \frac{1}{\tan \theta}\) and \(\csc \theta = \frac{1}{\sin \theta}\). Also recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), where \(\theta\) is the angle in the trigonometric function.

Step 2: Rewrite the equation using identities

We transform the given expression \(\cot \theta = \sqrt{\csc^2 \theta + 1}\) using the formulas from Step 1: This gives \(\frac{1}{\tan \theta} = \sqrt{(\frac{1}{\sin \theta})^2 + 1}\)

Step 3: Simplify the equation further using trigonometric identities

Simplify the equation further: \(\frac{1}{\frac{\sin \theta}{\cos \theta}} = \sqrt{(\frac{1}{\sin^2 \theta}) + 1}\) This simplifies to \(\frac{\cos \theta}{\sin \theta} = \sqrt{\frac{1 + \sin^2 \theta}{\sin^2 \theta}}\). Thus the equation simplifies to \(\frac{\cos \theta}{\sin \theta} = \frac{\sqrt{1 + \sin^2 \theta}}{\sin \theta}\).

Step 4: Check if both sides of the equation are equal

At this point, we can see that both sides of the equations are not equal, hence the given equation is not an identity.

Key Concepts

Understanding Cotangent

The cotangent of an angle \(\theta\) is a fundamental trigonometric function that complements the tangent function. It is defined as the reciprocal of the tangent, so \(\cot \theta = \frac{1}{\tan \theta}\). This means \(\cot \theta\) is essentially the ratio of the adjacent side over the opposite side in a right triangle. Or mathematically, \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). This function helps in solving various trigonometric equations, especially where the tangent function is involved.
Understanding relationships like \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) is important. It aids in simplifying expressions that involve both sine and cosine. These relationships are vital in verifying identities and solving trigonometric equations. Remember, recognizing these reciprocal and ratio relationships is key to mastering trigonometry.

Exploring Cosecant

The cosecant of an angle \(\theta\), denoted as \(\csc \theta\), is another reciprocal trigonometric function. It is defined as the reciprocal of the sine function, thus \(\csc \theta = \frac{1}{\sin \theta}\). This means when sine has a small value, \(\csc \theta\) becomes larger and vice versa.
In terms of right-angled triangles, \(\csc \theta\) represents the ratio of the hypotenuse over the opposite side. This can be particularly useful in solving equations where the sine function gets unwieldy and difficult to work with. In our exercise, it was used in the equation \(\sqrt{\csc^2 \theta + 1}\), demonstrating its utility in a complex expression. Understanding how to manipulate this function using identities can make solving trigonometric problems easier. Keep in mind the operations and transformations involved when you encounter expressions with \(\csc \theta\).

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions often involves using identities and relationships among functions. These simplifications can convert complex equations into more manageable forms. In our exercise, the transformation started with the equation \(\cot \theta = \sqrt{\csc^2 \theta + 1}\), which was rewritten to involve basic trigonometric identities.

• Use reciprocal identities: such as \(\cot \theta = \frac{1}{\tan \theta}\) for simplification.
• Simplify square roots in expressions: recalibrate complex expression terms using identities like \(\csc \theta = \frac{1}{\sin \theta}\).
• Compare and verify both sides of the equation: the output may show if the original equation holds as an identity or not.

In this challenge, by simplifying step-by-step, we determined the equation did not hold as an identity. Always cross-check every transformation, ensuring all operations adhere to trigonometric principles. Knowing how to break down each part is the heart of solving these expressions efficiently.