Question

In problems 19-98, establish each identity.

$\frac{1-\sin \left(v\right)}{\cos \left(v\right)}+\frac{\cos \left(v\right)}{1-\sin \left(v\right)}=2\sec \left(v\right)$

Short answer

We have proved the identity$\frac{1-\sin \left(v\right)}{\cos \left(v\right)}+\frac{\cos \left(v\right)}{1-\sin \left(v\right)}=2\sec \left(v\right)$.

Step-by-step solution

Step 1. Given information.

We have:

$\frac{1-\sin \left(v\right)}{\cos \left(v\right)}+\frac{\cos \left(v\right)}{1-\sin \left(v\right)}=2\sec \left(v\right)$

Step 2. Establish the identity. 

Take the left hand side:

$\frac{1-\sin \left(v\right)}{\cos \left(v\right)}+\frac{\cos \left(v\right)}{1-\sin \left(v\right)}$

$$\begin{gathered} =\frac{{\left(1-\sin \left(v\right)\right)}^2+{\cos }^2\left(v\right)}{\left(1-\sin \left(v\right)\right)\cos \left(v\right)} \\ =\frac{1+{\sin }^2\left(v\right)-2\sin \left(v\right)+{\cos }^2\left(v\right)}{\left(1-\sin \left(v\right)\right)\cos \left(v\right)} \\ =\frac{2-2\sin \left(v\right)}{\left(1-\sin \left(v\right)\right)\cos \left(v\right)} \\ =\frac{2\left(1-\sin \left(v\right)\right)}{\left(1-\sin \left(v\right)\right)\cos \left(v\right)} \\ =\frac{2}{\cos \left(v\right)} \\ =2\sec \left(v\right) \end{gathered}$$

Hence, proved.