Problem 1 - Algebra 2

Question

Verify each identity. $$\cos \theta \cot \theta=\frac{1}{\sin \theta}-\sin \theta$$

Step-by-Step Solution

Answer

The identity \(\cos \theta \cot \theta = \frac{1}{\sin \theta} - \sin \theta\) is indeed correct.

Step 1: Write cotangent in terms of sine and cosine

The left hand side (LHS) of the equation is \( \cos \theta \cot \theta \). To simplify this, write cotangent in terms of sine and cosine: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). This then gives \( \cos \theta \cdot \frac{\cos \theta}{\sin \theta} = \frac{\cos^2 \theta}{\sin \theta} \).

Step 2: Use Pythagorean Identity

Use the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \) to replace \( cos^2 \theta \) in the LHS. This results in \( \frac{1 - \sin^2 \theta}{\sin \theta} \) for the LHS.

Step 3: Simplify the expression

Now, separate the fraction to align with the RHS of the equation: \( \frac{1}{\sin \theta} - \sin \theta \).

Step 4: Compare LHS and RHS

Now the left hand side is equal to the right hand side. So the identity \( \cos \theta \cot \theta = \frac{1}{\sin \theta} - \sin \theta \) is verified.

Key Concepts

Understanding the Cosine Function

Cosine is one of the fundamental trigonometric functions, commonly denoted as \( \cos \theta \). It is derived from the ratio of the adjacent side to the hypotenuse in a right-angled triangle. If you imagine a circle drawn within the right triangle, cosine can also be seen as the horizontal coordinate or the x-value of a point on the unit circle.

Cosine has values ranging from -1 to +1, which repeat every 360 degrees or \( 2\pi \) radians.

• This periodic nature is crucial in simplifying trigonometric expressions and proving identities, such as the one in the exercise.
• In our specific example, recognizing \( \cos^2 \theta \) allowed us to use the Pythagorean Identity\( \cos^2 \theta + \sin^2 \theta = 1 \), which is a key identity in trigonometry.

By reorganizing cosine terms using identities, we simplify complex expressions, showing how trigonometric functions relate to one another.

Introducing the Sine Function

The sine function, denoted as \( \sin \theta \), is another primary trigonometric function that describes the ratio of the opposite side to the hypotenuse in a right triangle. Sine can also be visualized as the vertical coordinate, or the y-value, of a point on the unit circle.

Just like cosine, sine values oscillate between -1 and +1 and have the same periodicity, repeating every \( 2\pi \) radians.

• In trigonometric identities, sine plays a pivotal role, contributing to expressions alongside cosine.
• The Pythagorean Identity \( \cos^2 \theta + \sin^2 \theta = 1 \) links sine and cosine together, providing a powerful tool for transforming expressions.

In the given identity, sine is used on both sides of the equation. Transforming and simplifying formulas often involve expressing one trigonometric function in terms of another, as demonstrated here.

The Cotangent Function Unveiled

The cotangent function \( \cot \theta \) is defined as the reciprocal of the tangent function. It is expressed as \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).

This makes cotangent particularly useful when dealing with expressions involving both sine and cosine.

• In trigonometry, understanding reciprocals and their properties can significantly simplify solving identities.
• For instance, in our exercise, expressing cotangent in terms of sine and cosine allowed the identity verification by simplifying each side.

By substituting cotangent with its equivalent expressions in terms of sine and cosine, we ease comparison between different parts of an identity, emphasizing the interconnectedness of trigonometric functions.