Trigonometry
We are given the equation:
[tex]\tan (x)\csc (x)=\frac{1}{f(x)}[/tex]
It's required to write f(x) in terms of the sine and cosine functions.
Taking the reciprocal of both sides of the equation:
[tex]f(x)=\frac{1}{\tan (x)\csc (x)}[/tex]
Recall:
[tex]\begin{gathered} \tan (x)=\frac{\sin (x)}{\cos (x)} \\ \text{csc(x)}=\frac{1}{\sin (x)} \end{gathered}[/tex]
Substituting:
[tex]f(x)=\frac{1}{\frac{\sin(x)}{\cos(x)}\frac{1}{\sin (x)}}[/tex]
Simplifying:
[tex]f(x)=\frac{1}{\frac{1}{\cos(x)}}=\cos (x)[/tex]
Thus:
f(x)= cos(x)
