Problem 70 - Calculus for Biology and Medicine

Question

Show that the identity \(1+\cot ^{2} \theta=\csc ^{2} \theta\) follows from \(\sin ^{2} \theta+\cos ^{2} \theta=1\)

Step-by-Step Solution

Answer

Pythagorean identity transforms into the desired identity by substitution and simplification.

Step 1: Understand the trigonometric identities

We begin with the Pythagorean identity: \[sin^2 \theta + cos^2 \theta = 1.\]We need to show that it leads to the identity:\[1 + \cot^2 \theta = \csc^2 \theta.\]

Step 2: Express terms in terms of sin and cos

Recall that the cotangent and cosecant functions are defined as follows:\[\cot \theta = \frac{\cos \theta}{\sin \theta},\quad \csc \theta = \frac{1}{\sin \theta}.\]Thus, \[\cot^2 \theta = \left(\frac{\cos \theta}{\sin \theta}\right)^2 = \frac{\cos^2 \theta}{\sin^2 \theta},\]\[\csc^2 \theta = \left(\frac{1}{\sin \theta}\right)^2 = \frac{1}{\sin^2 \theta}.\]

Step 3: Substitute and manipulate back to identity

Substitute these forms into the identity we want to prove:\[1 + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta}.\]Combine the terms on the left-hand side into a single fraction:\[\frac{\sin^2 \theta}{\sin^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta}.\]Since \( \sin^2 \theta + \cos^2 \theta = 1 \), the expression on the left simplifies to:\[\frac{1}{\sin^2 \theta}.\]Observe that both sides are equal, proving the identity.

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the foundational concepts in trigonometry. It states that for any angle \( \theta \), the sum of the squares of sine and cosine is always 1. This is mathematically expressed as \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is not just a random equation — it stems from the Pythagorean Theorem, applied to the unit circle (a circle with radius 1).
When you take a point on the unit circle, its coordinates are \((\cos \theta, \sin \theta)\). According to the Pythagorean Theorem, the square of the length of the hypotenuse (which is the radius of the circle, and thus 1) is equal to the sum of the squares of the other two sides, which are \( \cos \theta \) and \( \sin \theta \). This gives us the identity.

• Useful for proving other identities.
• Forms the basis of trigonometry on the unit circle.
• Helps in converting between trigonometric functions.

Mastering the Pythagorean identity is crucial because it serves as a key stepping stone in understanding more complex trigonometric relationships.

Cotangent Function

The cotangent function, denoted as \( \cot \theta \), is one of the six fundamental trigonometric functions. It is the reciprocal of the tangent function and is defined as the ratio of the cosine to the sine of an angle: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Because of its definition, the cotangent function highlights the relationship between the adjacent and opposite sides of a right triangle.

Here are some key points about the cotangent function:

• The cotangent function is undefined whenever sine is zero, which occurs at integer multiples of \( \pi \).
• In terms of angle of elevation or depression, \( \cot \theta \) provides the ratio of the horizontal distance to the vertical height.
• This function is useful in various mathematical and engineering fields, especially when dealing with periodic phenomena.

Understanding cotangent is vital for solving problems that involve angles and their measure, especially in calculus and trigonometry courses.

Cosecant Function

The cosecant function, symbolized as \( \csc \theta \), is another essential trigonometric function. It is the reciprocal of the sine function and is represented as \( \csc \theta = \frac{1}{\sin \theta} \). Because it's the reciprocal, it helps when sine values are very small (close to zero), making \( \csc \theta \) extremely large or even undefined.

Important aspects of the cosecant function include:

• The cosecant function is undefined whenever \( \sin \theta = 0 \), which happens at integer multiples of \( \pi \).
• It is particularly useful in mastering the Pythagorean identity transformation to prove other identities, like in the given exercise where \( 1 + \cot^2 \theta = \csc^2 \theta \) stems from the Pythagorean identity.
• Cosecant can provide insights into the relationship between lengths in triangles or circular motion.

Mastery over the cosecant function is necessary when dealing with various trigonometric identities and transformations. It ensures a comprehensive understanding of deeper mathematical concepts.