Trig Identities Exercise 1

Answer: \(\tan x\)

Solution Method:The Reciprocal and Ratio Identities are the most basic identities. The three reciprocal identities simply define the reciprocal functions. Then the ratio identities define tangent and cotangent in terms of sine and cosine. 
  \begin{array}{cc}
 \csc{x}=\dfrac{1}{\sin{x}} & \tan{x} = \dfrac{\sin{x}}{\cos{x}} \\[8pt]
 \sec{x}=\dfrac{1}{\cos{x}} &    \cot{x} = \dfrac{\cos{x}}{\sin{x}}\\[8pt]
   \cot{x}=\dfrac{1}{\tan{x}}
\end{array}
When simplifying an equation using trig identities, our goal is to get to a single function or number. A helpful trick is to get everything in terms of sine and cosine, because all trig functions can be defined by those two, as you see from these identities. 
\begin{equation*}
    \frac{\sin^2{x}}{\cos{x}} \cdot \csc{x} =
    \frac{\sin^2{x}}{\cos{x}} \cdot \frac{1}{\sin{x}} =
    \frac{\sin{x}}{\cos{x}} =
    \tan{x}
\end{equation*}