Problem 39 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ (\csc t-\cot t)^{2}=\frac{1-\cos t}{1+\cos t} $$

Step-by-Step Solution

Answer

The identity \((\csc t - \cot t)^{2} = \frac{1 - \cos t}{1 + \cos t} \) is verified as true by simplification.

Step 1: Expand the Left Side

Start by expanding the expression \((\csc t - \cot t)^2\). Recall that: \((a-b)^2 = a^2 - 2ab + b^2\). Therefore, use this identity for the expansion. Here, \(a = \csc t\) and \(b = \cot t\). So it becomes: \[ (\csc t)^2 - 2(\csc t)(\cot t) + (\cot t)^2 \]

Step 2: Simplify Using Trig Identities

Recall the trigonometric identities: \(\csc t = \frac{1}{\sin t}\) and \(\cot t = \frac{\cos t}{\sin t}\). Use these identities to further express the expanded terms:1. \( (\csc t)^2 = \frac{1}{\sin^2 t} \)2. \( 2(\csc t)(\cot t) = 2 \times \frac{1}{\sin t} \times \frac{\cos t}{\sin t} = \frac{2 \cos t}{\sin^2 t} \)3. \( (\cot t)^2 = \left( \frac{\cos t}{\sin t} \right)^2 = \frac{\cos^2 t}{\sin^2 t} \)Combine the terms: \[ \frac{1}{\sin^2 t} - \frac{2 \cos t}{\sin^2 t} + \frac{\cos^2 t}{\sin^2 t} = \frac{1 - 2\cos t + \cos^2 t}{\sin^2 t} \]

Step 3: Relate to the Identity \(1 - \cos t\) and \(1 + \cos t\)

Consider the expression on the right side: \(\frac{1 - \cos t}{1 + \cos t}\). Use the identity: \(\sin^2 t = 1 - \cos^2 t\). Therefore, adjust the left side's numerator:Recall that \((1 - \cos t)^2 = 1 - 2\cos t + \cos^2 t\), which matches the numerator \(1 - 2\cos t + \cos^2 t\). The denominator \(\sin^2 t\) is equivalent to \(1 - \cos^2 t\), but simplifying \(\frac{(1 - \cos t)^2}{\sin^2 t}\) leads to \(\frac{1 - \cos t}{1 + \cos t}\).

Step 4: Verify Final Equality

Having shown that both sides simplify to the same expression:The left side becomes:\[ \frac{(1 - \cos t)^2}{\sin^2 t} \] and using the identity \(\sin^2 t = (1-\cos t)(1+\cos t)\), it simplifies to:\[ \frac{1 - \cos t}{1 + \cos t} \]Thus, the original identity is verified.

Key Concepts

Cosecant

The cosecant is an often misunderstood member of the trigonometric family. Known as \(\csc t\), it is the reciprocal of the sine function. Essentially, this means that \(\csc t = \frac{1}{\sin t}\). If you are dealing with angles in a right triangle, think of cosecant as the ratio of the hypotenuse to the opposite side.
Understanding this function is key to simplifying many trigonometric identities and equations. Whenever you encounter \(\csc t\), remember to switch to \(\frac{1}{\sin t}\) for easier calculations.

Cotangent

Another crucial trigonometric function is the cotangent, denoted as \(\cot t\). This is the reciprocal of the tangent function, which expresses the ratio of the adjacent side over the opposite side in a right triangle. Its formula is \(\cot t = \frac{\cos t}{\sin t}\).
The cotangent plays a significant role in trigonometric simplifications. By expressing cotangent in terms of sine and cosine, you create an opportunity to simplify complex expressions and verify trigonometric identities more easily.

Trigonometric Simplification

Trigonometric simplification involves breaking down complex trigonometric expressions into their simplest forms. This process utilizes basic trigonometric identities and reciprocal identities such as \(\csc t\) and \(\cot t\).
Here's a useful approach:

• Rewrite functions using trigonometric identities (like \(\csc t = \frac{1}{\sin t}\) and \(\cot t = \frac{\cos t}{\sin t}\)).


• Look for common factors to simplify terms.


• Use algebraic techniques such as factoring and canceling common denominators.

Simplification makes solving and verifying trigonometric equations easier, turning a seemingly complex task into a straight-forward one.

Pythagorean Identity

The Pythagorean identity is one of the cornerstone identities in trigonometry. Its basic form is \(\sin^2 t + \cos^2 t = 1\). This identity is incredibly useful for converting between trigonometric functions and simplifying expressions.
In this exercise, another form, \(\sin^2 t = 1 - \cos^2 t\), plays a critical role. When simplifying or verifying identities like the original problem, this transformation helps relate expressions with \(\sin\) and \(\cos\) directly to each other.
Understanding this identity allows you to pivot between trigonometric expressions easily, making problem-solving much more manageable.