We are given the following trigonometric expression:
[tex]\frac{\sin \theta-\csc \theta}{\cos \theta}[/tex]
We are asked to put this expression as a function of sines and cosines, t do this, let's remember the following relationship:
[tex]\csc \theta=\frac{1}{\sin \theta}[/tex]
Now we replace this in the expression:
[tex]\frac{\sin \theta-\frac{1}{\sin\theta}}{\cos \theta}[/tex]
Now we do the operation in the numerator, like this:
[tex]\frac{\frac{(\sin\theta)(\sin\theta)-1}{\sin\theta}}{\cos \theta}[/tex]
We simplify the result:
[tex]\frac{\sin ^2\theta-1}{\sin \theta\cos \theta}[/tex]
now we use the following relationship:
[tex]\begin{gathered} \sin ^2\theta+\cos ^2\theta=1 \\ \cos ^2\theta=1-\sin ^2\theta \\ -\cos ^2\theta=\sin ^2\theta-1 \end{gathered}[/tex]
replacing this in the expression
[tex]\frac{-\cos ^2\theta}{\sin \theta\cos \theta}[/tex]
simplifying:
[tex]-\frac{\cos \theta}{\sin \theta}[/tex]
Since we are asked to put the expression as a function of sines and cosines we can't simplify any further, therefore the previous expression is the result.
