Since cosecant is a reciprocal of sine, it uses the same general formula of the sine function with the letters corresponding to the same transformations.  Note that while A does correspond to amplitude, the cosecant function extends infinitely upwards and downwards so there is no amplitude for the graphs.

Knowing that the graph of  is

we can use the general form of the cosecant transformation equation, , and apply these transformations.  We can ignore  because in this case .  In this case  and so our period is: 

Period = 

Period = 

Period = 

 is the normal period for cosecant graphs and so we do not have to worry about lengthening or shortening the period.   and so we need to apply a phase shift of .  This will cause our graph to shift left a total of  units.

Lastly, we must apply the transformation for , so we will have an upward vertical shift of 1 unit.

The application of these transformations leaves us with our graph of  . 

 In order to understand the graph of secant, recall that .  First consider the graph of .

Anywhere this graph crosses the x-axis a vertical asymptote will form for the  graph because the denominator of  will be equal to zero and the function will be undefined.  At each maximum and minimum of , the graph of  will invert at that point.

And then we are left with the graph . 

When looking at the graph of , it extends infinitely upwards and downwards from each local maximum and minimum.  This will be true for all transformed secant graphs as well.  Due to this, there is no amplitude for secant graphs.  However, secant is the reciprocal of cosine graphs which do rely on amplitude for transformations.  For this reason amplitude must be considered as a vertical shift.

Knowing that the general form of the graph  is:

We can use the general form of the cosecant transformation equation, , and apply these transformations.   because secant graphs extend infinitely upwards and downwards and does not have an amplitude, we must think of the secant graph being a reciprocal of the cosine graph.  So we will consider  for cosine.

We will shift our secant graph to invert at the maximums and minimums of the cosine graph.

Next, we will factor  in order to get our equation into the form .

And so .  We can now solve for our period,

Period = 

Period = 

Period = 

This shortens our original period of  to .

Now we must consider .  This will give us a phase shift of  units to the left.  Since our period has also been shortened this does not change the graph visually. in this case so we do not need to consider a vertical shift.

And we are left with the graph of .

In trigonometry,

.

It may also be thought of as .

This is because

and , so

. 

This means that whenever cosine is 0, tangent is undefined, because it would be evaluated by dividing by 0.

We can factor the original expression as follows:

So from this equation we conclude either that:

or

So any number that is not some integer multiple of  away from these two solutions is not a solution to the original equation.

The only such choice is  , which is  ; n is not an integer, therefore it is not a solution.

The graph looks to have infinite range, but multiple vertical asymptotes. That means we can limit our choices to tangent and cotangent graphs.

Furthermore, we observe that the graph starts at the bottom and increases from left to right, consistent with tangent graphs. So we narrow our focus to the choices involving tangents.

To decide between the remaining two graphs, observe that y-intercept (where x=0) of our graph is (0,1).

Now   evaluated at  is , which means that we need a vertical shift of  unit.

Hence the best choice is:

To derive the graph of  , recall that .  The graph of   is

and the graph of is

Vertical asymptotes will occur in the graph of   whenever .  This is because the denominator of the tangent function will be equal to zero whenever the cosine function is equal to zero and then the entire function will be undefined at those points.  Wherever cosine crosses the x-axis a vertical asymptote will occur.  If we overlay the sine and cosine graphs we see the following:

So our tangent graph will follow the same form as the sine and cosine graphs when they are increasing, but will have vertical asymptotes wherever cosine crosses the x-axis.

And we are left with our graph of 

We will begin by considering the general graph of  and apply transformations step by step to produce a graph of .  The graph of  is

The general equal of a tangent transformation equation is .  A is the amplitude of the graph of tangent.  Here,  so we do not need to apply a transformation here.  Next, we will consider the period.  The period of the tangent function is equal to .  So the period of our graph would be

Period = 

Period = 

So the period is shortened from  to .

Now, we will consider .   is the phase shift of our graph.  So we will shift our graph  units to the left.  This does not change our graph visually due to the period now being .  Lastly, we will consider .   is the vertical shift of our graph, and so we must shift our graph 1 unit up.

And we are left with the graph of .