Write the following in terms of \( \sin (x) \) and \( \cos (x) \), and then simplify if possible. Leave your answer in terms of sines and cosines only. \( \csc (x) \tan (x)=\square \)

To express \( \csc(x) \tan(x) \) in terms of \( \sin(x) \) and \( \cos(x) \), we can start by recalling the definitions of the cosecant and tangent functions: \[ \csc(x) = \frac{1}{\sin(x)} \quad \text{and} \quad \tan(x) = \frac{\sin(x)}{\cos(x)}. \] Now we can substitute these definitions into the expression: \[ \csc(x) \tan(x) = \frac{1}{\sin(x)} \cdot \frac{\sin(x)}{\cos(x)}. \] When we multiply these two fractions, the \( \sin(x) \) in the numerator and denominator cancels out: \[ = \frac{1}{\cos(x)}. \] Thus, the expression simplifies to: \[ \csc(x) \tan(x) = \sec(x). \] However, if we want to express it as just sine and cosine, we keep it as: \[ \csc(x) \tan(x) = \frac{1}{\cos(x)}. \] So we have: \[ \csc(x) \tan(x) = \frac{1}{\cos(x)} = \sec(x). \] If you were looking for the answer strictly in terms of sine and cosine, the final expression we need is: \[ \csc(x) \tan(x) = \frac{1}{\cos(x)}. \]