Problem 19 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \dfrac{\cot^2 t}{\csc t} = \dfrac{1 - \sin^2 t}{\sin t} \)

Step-by-Step Solution

Answer

The given identity \( \dfrac{\cot^2 t}{\csc t} = \dfrac{1 - \sin^2 t}{\sin t} \) is verified.

Step 1: Understand the basic identities

Understand that cotangent (cot) is the reciprocal of tangent (tan), or \( \cot t = \dfrac{1}{\tan t} = \dfrac{\cos t}{\sin t} \) and that cosecant (csc) is the reciprocal of sine (sin), or \( \csc t = \dfrac{1}{\sin t} \). So, we can rewrite \( \cot^2 t \) as \( \left(\dfrac{\cos t}{\sin t}\right)^2 \) and \( \csc t \) as \( \dfrac{1}{\sin t} \).

Step 2: Simplify left side of the equation

Substitute the values from Step 1 into the left side of the equation \( \dfrac{\cot^2 t}{\csc t} \). This gives us \( \dfrac{\left(\dfrac{\cos^2 t}{\sin^2 t}\right)}{\dfrac{1}{\sin t}} \). Simplifying this, we get \( \dfrac{\cos^2 t}{\sin t} \).

Step 3: Use Pythagorean identity

We know from the Pythagorean identity that \( \cos^2 t = 1 - \sin^2 t \). So, \( \dfrac{\cos^2 t}{\sin t} \) can be rewritten as \( \dfrac{1 - \sin^2 t}{\sin t} \).

Step 4: Compare both sides of the equation

Now, both sides of the equation \( \dfrac{\cot^2 t}{\csc t} = \dfrac{1 - \sin^2 t}{\sin t} \) are identical, which verifies the identity.

Key Concepts

Cotangent

The cotangent is a fundamental trigonometric concept often abbreviated as "cot." It is defined for an angle in a right triangle and is the reciprocal of the tangent function. The formula for cotangent is:

• \( \cot t = \dfrac{1}{\tan t} = \dfrac{\cos t}{\sin t} \)

In other words, cotangent is the ratio of the adjacent side to the opposite side in a right triangle.
Understanding this makes it easier to manipulate expressions involving cotangent, especially when verifying identities.
For instance, when you square the cotangent, you essentially square both the numerator and the denominator, which leads to:

• \( \cot^2 t = \left(\dfrac{\cos t}{\sin t}\right)^2 = \dfrac{\cos^2 t}{\sin^2 t} \)

This can be useful in verifying trigonometric identities as it allows for substitution and simplification of expressions.

Cosecant

Cosecant, abbreviated as "csc," is another reciprocal trigonometric function and is the counterpart to the sine function. The formula for cosecant is:

• \( \csc t = \dfrac{1}{\sin t} \)

This means that cosecant is the reciprocal of sine for any angle, which is equivalent to the hypotenuse over the opposite side in a right triangle context.
Understanding how cosecant works is crucial when simplifying trigonometric expressions. For example, knowing that \( \csc t \) can be replaced with \( \dfrac{1}{\sin t} \) allows you to simplify expressions by eliminating sine, which may be beneficial depending on the identity you need to verify.
In this exercise, you can see cosecant being replaced and rearranged within the expression to test and confirm the given identity.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental identities in trigonometry and relates the sine and cosine of an angle. It is expressed as:

• \( \sin^2 t + \cos^2 t = 1 \)

This identity can be altered to express either sine or cosine in terms of the other:

• \( \cos^2 t = 1 - \sin^2 t \)
• \( \sin^2 t = 1 - \cos^2 t \)

Using these relationships allows for substitutions that simplify trigonometric identities and prove different equations.
For instance, in this exercise, using \( \cos^2 t = 1 - \sin^2 t \) simplifies the expression into a form that can easily be verified as equal on both sides of the given identity.
Recognizing and applying the Pythagorean identity enables students to solve a wide array of trigonometric problems efficiently and with confidence.