Answer
See the full explanation below.
Work Step by Step
$\frac{\csc x-\sec x}{\csc x+\sec x}$
Apply the reciprocal identity of trigonometry, $\csc x=\frac{1}{sinx}$ , and $\sec x=\frac{1}{\cos x}$. Then the above expression can be further simplified as:
$\frac{\csc x-\sec x}{\csc x+\sec x}=\frac{\frac{1}{\sin x}-\frac{1}{\cos x}}{\frac{1}{\sin x}+\frac{1}{\cos x}}$
We multiply the numerator and denominator by $\cos x$.
$\begin{align}
& \frac{\frac{1}{\sin x}-\frac{1}{\cos x}}{\frac{1}{\sin x}+\frac{1}{\cos x}}=\frac{\frac{1}{\sin x}-\frac{1}{\cos x}}{\frac{1}{\sin x}+\frac{1}{\cos x}}.\frac{\cos x}{\cos x} \\
& =\frac{\frac{\cos x}{\sin x}-\frac{\cos x}{\cos x}}{\frac{\cos x}{\sin x}+\frac{\cos x}{\cos x}} \\
& =\frac{\frac{\cos x}{\sin x}-1}{\frac{\cos x}{\sin x}+1}
\end{align}$
By using the quotient identity of trigonometry $\cot x=\frac{\cos x}{\sin x}$, the above expression can be further simplified as:
$\frac{\frac{\cos x}{\sin x}-1}{\frac{\cos x}{\sin x}+1}=\frac{\cot x-1}{\cot x+1}$
Thus, the left side of the expression is equal to the right side, which is
$\frac{\csc x-\sec x}{\csc x+\sec x}=\frac{\cot x-1}{\cot x+1}$.
