Problem 46 - A Graphical Approach to Precalculus with Limits

Question

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\frac{1}{1+\cot ^{2} \theta}$$

Step-by-Step Solution

Answer

The simplified expression is \( \sin^2 \theta \).

Step 1: Understand the Cotangent Identity

Recall that \( \cot \theta = \frac{1}{\tan \theta} \), which can be written in terms of sine and cosine as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Therefore, \( \cot^2 \theta = \left( \frac{\cos \theta}{\sin \theta} \right)^2 = \frac{\cos^2 \theta}{\sin^2 \theta} \).

Step 2: Use the Pythagorean Identity

The Pythagorean identity states that \( 1 + \tan^2 \theta = \sec^2 \theta \) or alternatively, using division, that \( \tan^2 \theta + 1 = \sec^2 \theta \). We are rewriting in terms of \( \cot \theta \) therefore the identity \( 1 + \cot^2 \theta = \csc^2 \theta \) (since \( \csc \theta = \frac{1}{\sin \theta} \)).Using this identity, \( 1 + \cot^2 \theta = \frac{1}{\sin^2 \theta} \).

Step 3: Substitute the Identity into the Expression

Substitute the result from Step 2 into the given expression:\[ \frac{1}{1 + \cot^2 \theta} = \frac{1}{\csc^2 \theta} = \sin^2 \theta \].

Step 4: Final Step: Simplify

Thus, the simplified expression in terms of sine is \( \sin^2 \theta \). Since there is no further simplification possible beyond this point, the expression can remain as \( \sin^2 \theta \).

Key Concepts

Cotangent

The cotangent function is one of the fundamental trigonometric functions, and it is closely connected to the tangent function. While the tangent of an angle \( \theta \) is the ratio of the sine to the cosine, the cotangent is its reciprocal. In mathematical terms, the cotangent of \( \theta \) is expressed as:

• \( \cot \theta = \frac{1}{\tan \theta} \)
• This can also be written in terms of sine and cosine as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

Understanding cotangent is crucial as it helps to convert complex trigonometric expressions into simpler terms using sine and cosine. This is particularly useful because many identities and trigonometric transformations rely on these two basic functions. In our exercise, the expression \( \cot^2 \theta \) appears, and rewriting it as \( \frac{\cos^2 \theta}{\sin^2 \theta} \) helps to simplify the problem further.

Pythagorean Identity

The Pythagorean Identity is a key cornerstone in trigonometry, often used to simplify expressions. It stems from the Pythagorean theorem that applies to right-angled triangles but is adapted for trigonometric purposes. The identity in its most common form is:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

There are variations of this identity that involve other trigonometric functions. For instance, when dealing with the cotangent, the identity \( 1 + \cot^2 \theta = \csc^2 \theta \) is extremely useful. Here's why this is important:

• \( \csc \theta \), the cosecant, is the reciprocal of sine, expressed as \( \csc \theta = \frac{1}{\sin \theta} \)

This identity allows us to transform the expression \( 1 + \cot^2 \theta \) into \( \frac{1}{\sin^2 \theta} \), effectively making the simplification process more straightforward. The key benefit is that it converts more complex trigonometric terms into the basic sine and cosine, which are easier to handle.

Sine and Cosine

Sine and cosine are the foundational pillars of trigonometry. Understanding these functions is vital for solving a wide range of trigonometric problems. Here’s a brief overview:

• Sine: Represented as \( \sin \theta \), it is the opposite side over the hypotenuse in a right triangle.
• Cosine: Denoted as \( \cos \theta \), it is the ratio of the adjacent side to the hypotenuse.

These functions serve as building blocks for all other trigonometric functions and identities. Simplifying trigonometric expressions often involves reducing them into terms of sine and cosine. For example, when given the expression \( \frac{1}{1 + \cot^2 \theta} \), the key to simplification lies in expressing everything in terms of sine and cosine:

• Express \( \cot^2 \theta \) as \( \frac{\cos^2 \theta}{\sin^2 \theta} \)
• Utilize the identity to transform \( 1 + \cot^2 \theta \) into \( \csc^2 \theta \)
• The final simplification results in \( \sin^2 \theta \)

This example highlights the power of converting expressions to sine and cosine to both simplify and better understand the problem.