Answer
$$\tan x(\cot x+\csc x)=1+\sec x$$
Work Step by Step
$$A=\tan x(\cot x+\csc x)$$
$$A=\tan x\cot x+\tan x\csc x$$
- Reciprocal Identity:
$$\cot x=\frac{1}{\tan x}$$
Therefore, $$\tan x\cot x=1$$
Also, from Reciprocal Identity: $$\csc x=\frac{1}{\sin x}$$
and Quotient Identity: $$\tan x=\frac{\sin x}{\cos x}$$
We can make out that $$\tan x\csc x=\frac{\sin x}{\cos x}\times\frac{1}{\sin x}=\frac{1}{\cos x}=\sec x\hspace{1cm}\text{(Reciprocal Identity)}$$
Overall, we can now apply these results to $A$, $$A=1+\sec x$$
