Chapter 7, Problem 54RE

To determine

To establish:the given identity.

Answer to Problem 54RE

The identity is established.

Explanation of Solution

Given:

The given identity is $\frac{\csc \theta }{1-\cos \theta }=\frac{1+\cos \theta }{{\sin }^3\theta }$ .

Calculation:

As the given identity is $\frac{\csc \theta }{1-\cos \theta }=\frac{1+\cos \theta }{{\sin }^3\theta }$ .

Now, let us check LHS:

  $\begin{array}{l}LHS=\frac{\csc \theta }{1-\cos \theta }\\ \{\text{multiply by its cojugate}}\\ =\frac{\csc \theta }{\left(1-\cos \theta \right)}\times \frac{\left(1+\cos \theta \right)}{\left(1+\cos \theta \right)}\\ =\frac{\csc \theta \left(1+\cos \theta \right)}{\left(1-{\cos }^2\theta \right)}\\ =\frac{\csc \theta \left(1+\cos \theta \right)}{{\sin }^2\theta }\\ =\frac{\left(1+\cos \theta \right)}{sin\theta {\sin }^2\theta }\\ =\frac{1+\cos \theta }{{\sin }^3\theta }\\ =RHS\end{array}$

Thus, $\frac{\csc \theta }{1-\cos \theta }=\frac{1+\cos \theta }{{\sin }^3\theta }$

Hence, the identity is established.