Chapter 5.2, Problem 22E

To determine

To verify the identity $\frac{\cos \theta \cot \theta }{1-\sin \theta }-1=\csc \theta$ algebraically and to use table feature of graphing utility to check the result numerically

Answer to Problem 22E

True

Explanation of Solution

Given:

Equation: $\frac{\cos \theta \cot \theta }{1-\sin \theta }-1=\csc \theta$

Formula used:

  $\begin{array}{l}\cos \theta =\frac{\cos \theta }{\sin \theta }\\ \sin \theta =\frac{1}{sin\theta }\end{array}$

Calculation:

1. Algebraic method

Consider Left Hand side of the given equation,

  $\begin{array}{l}\Rightarrow \frac{\cos \theta \cot \theta }{1-\sin \theta }-1\\ \Rightarrow \frac{\cos \theta \left(\frac{\cos \theta }{\sin \theta }\right)}{1-\sin \theta }-1[\because \cos \theta =\frac{\cos \theta }{\sin \theta }]\\ \Rightarrow \frac{{\cos }^2\theta }{\sin \theta \left(1-\sin \theta \right)}-1\\ \Rightarrow \frac{{\cos }^2\theta -\sin \theta \left(1-\sin \theta \right)}{\sin \theta \left(1-\sin \theta \right)}\end{array}$

  $\begin{array}{l}\Rightarrow \frac{{\cos }^2\theta -\sin \theta \left(1-\sin \theta \right)}{\sin \theta \left(1-\sin \theta \right)}[\because \text{Taking Least Common Factor}]\\ \Rightarrow \frac{{\cos }^2\theta -\sin \theta +{\sin }^2\theta }{\sin \theta \left(1-\sin \theta \right)}\\ \Rightarrow \frac{1-\sin \theta }{\sin \theta \left(1-\sin \theta \right)}\\ \Rightarrow \frac{1}{sin\theta }\\ \Rightarrow \csc \theta [\because \sin \theta =\frac{1}{sin\theta }]\\ \Rightarrow \csc \theta \mathrm{......}(i)\end{array}$

Now, consider Right Hand Side of the given equation,

  $\Rightarrow \csc \theta \mathrm{......}(ii)$

It is clear that, equation (i) and equation (ii) are equal.

Therefore, Left Hand Side is equal to Right Hand Side.

Hence, $\frac{\cos \theta \cot \theta }{1-\sin \theta }-1=\csc \theta$ .

2. Numerical method

Let, $f(\theta )=\frac{\cos \theta \cot \theta }{1-\sin \theta }-1\text{and}g(\theta )=\csc \theta$ .

Let $\theta =x.$

Now, using the table feature of graphing utility,

  

Interpretation:

From the above table, it is clear that, for any value of x, both functions were resulting same value.

So, both functions are equal to each other.

Conclusion:

Therefore, $\frac{\cos \theta \cot \theta }{1-\sin \theta }-1=\csc \theta$ .