Problem 116 - Precalculus : Building Concepts and Connections

Question

Derive the Pythagorean identity \(1+\cot ^{2} t=\csc ^{2} t\)

Step-by-Step Solution

Answer

The Pythagorean identity \(1+\cot^{2}t=\csc^{2}t\) is confirmed through the steps involving the definitions of cotangent and cosecant and the standard Pythagorean identity.

Step 1: Definitions of Cotangent and Cosecant

Let's start by defining what cotangent and cosecant mean in trigonometric terms. By definition, \( \cot(t) = \frac{1}{\tan(t)} = \frac{\cos(t)}{\sin(t)} \) and \( \csc(t) = \frac{1}{\sin(t)} \) . So, we can replace the terms in the original identity with these definitions and rewrite the identity as: \( 1 + \left( \frac{\cos(t)}{\sin(t)} \right)^2 = \left( \frac{1}{\sin(t)} \right)^2 \)

Step 2: Expand the Square

Now we can simplify the right side of the equation by expanding the square. Upon expanding, the identity now becomes: \( 1 + \frac{\cos^2(t)}{\sin^2(t)} = \frac{1}{\sin^2(t)} \)

Step 3: Simplify the Identity

The next step will be to simplify the equation. We notice that \( 1/\sin^2(t) \) can be factored out of both terms on the left side of the equation. Factoring this term out gives us: \( \frac{1}{\sin^2(t)} * (1 + \cos ^{2}t )= \frac{1}{\sin^2(t)} \)

Step 4: Apply the Pythagorean Identity

Now we can apply the Pythagorean identity to simplify \( 1 + \cos^2(t) \) to \(\sin^2(t) + \cos^2(t)\) which is equal to 1 according to the Pythagorean identity. Now we have: \( \frac{1}{\sin^2(t)} * 1 = \frac{1}{\sin^2(t)} \)

Step 5: Final Simplification

Finally, we left with \( \frac{1}{\sin^2(t)} = \frac{1}{\sin^2(t)} \). That shows the original Pythagorean identity \(1+\cot^{2}t=\csc^{2}t\) is correct.

Key Concepts

Trigonometric Identities

Trigonometric identities form the backbone of understanding relationships between different trigonometric functions. These identities are essential tools for simplifying expressions and solving trigonometric equations. The Pythagorean identities, such as

• \( \sin^2(t) + \cos^2(t) = 1 \)
• \(1 + \tan^2(t) = \sec^2(t)\)
• \(1 + \cot^2(t) = \csc^2(t)\)

are used to express relationships involving squared trigonometric functions. These identities stem from the Pythagorean theorem in a right-angled triangle, where the hypotenuse squared equals the sum of the squares of the other two sides. Understanding these can aid in solving complex trigonometric problems and instantly recognizing equivalent expressions.

Cotangent

Cotangent, often symbolized as \( \cot(t) \), is the reciprocal of the tangent function. It is defined mathematically as:\[\cot(t) = \frac{1}{\tan(t)} = \frac{\cos(t)}{\sin(t)}\]This function represents the ratio of the adjacent side over the opposite side in a right-angled triangle.
Understanding cotangent is crucial for integrating and differentiating trigonometric functions, as well as simplifying trigonometric identities. In the Pythagorean identity \(1 + \cot^2(t) = \csc^2(t)\), it is used to relate the cotangent and cosecant functions, creating a bridge that links different trigonometric ratios and their reciprocals.

Cosecant

Cosecant, denoted as \( \csc(t) \), is the reciprocal of the sine function:\[\csc(t) = \frac{1}{\sin(t)}\]It represents the ratio of the hypotenuse to the opposite side in a right-angled triangle. Cosecant is less commonly used directly compared to sine or cosine but is invaluable when solving trigonometric identities and equations.
The identity \(1 + \cot^2(t) = \csc^2(t)\) illustrates how cosecant relates to other functions, showcasing its utility in expressing and verifying complex relationships within trigonometric contexts. Understanding cosecant aids in recognizing when to use certain identities and simplifies the solution process across various mathematical applications.