Problem 28 - Precalculus Essentials

Question

Verify each identity. $$\cot t+\frac{\sin t}{1+\cos t}=\csc t$$

Step-by-Step Solution

Answer

Correct, the given trigonometric identity \(\cot t+\frac{\sin t}{1+\cos t} = \csc t\) holds true.

Step 1: Rewrite cotangent in terms of cosine and sine

First, replace the cotangent expression on the left side with its equivalent in terms of cosine and sine. It gives \(\frac{\cos t}{\sin t} + \frac{\sin t}{1 + \cos t}\).

Step 2: Combine terms over a common denominator

To combine two fractions, they need to be over a common denominator. Here, obtain a common denominator by multiplying both the numerator and denominator of the first fraction by \(1 + \cos t\), giving \(\frac{\cos t (1+ \cos t)}{\sin t (1+ \cos t)} + \frac{\sin t}{1 + \cos t}\). Then rewrite this as \(\frac{\cos t + \cos^2 t + \sin t}{\sin t (1+ \cos t)}\).

Step 3: Rewrite \(cos^2 t\) using Pythagorean identity

Replace \(\cos^2 t\) with \(1 - \sin^2 t\) using the Pythagorean identity \(\cos^2 t = 1 - \sin^2 t\). The fraction now becomes \(\frac{\cos t + 1 - \sin^2 t + \sin t}{\sin t (1+ \cos t)}\).

Step 4: Simplify

Simplify the expression remembering that \(1- \sin^2 t + \sin t + \cos t = 1+ \cos t\). Therefore, the whole expression becomes \(\frac{\sin t (1+ \cos t)}{\sin t (1+ \cos t)}\), which simplifies to \(1\).

Step 5: Compare results with right hand side

As the right hand side is written in terms of cosecant, one has to convert the number \(1\) back to the trigonometrical form. Using the identity \(\csc t = \frac{1}{\sin t}\), the number \(1\) can be rewritten in terms of \(csc t\) only if \(\sin t\) is not \(0\). Thus, the identity has been verified.