Answer
See below.
Work Step by Step
b) In order to verify the equations, use the trigonometric identities.
$\tan x=\frac{\sin x}{\cos x}$ and $\cot x=\frac{\cos x}{\sin x}$
Now, apply the above identities in the following equation:
$\begin{align}
& \frac{\tan x-1}{1-\cot x}=\frac{\frac{\sin x}{\cos x}-1}{1-\frac{\cos x}{\sin x}} \\
& =\frac{\frac{\sin x-\cos x}{\cos x}}{\frac{\sin x-\cos x}{\sin x}} \\
& =\frac{\left( \sin x-\cos x \right)\sin x}{\left( \sin x-\cos x \right)\cos x} \\
& =\frac{\sin x}{\cos x}
\end{align}$
By using $\tan x=\frac{\sin x}{\cos x}$, we get:
$\begin{align}
& \frac{\tan x-1}{1-\cot x}=\frac{\sin x}{\cos x} \\
& =\tan x
\end{align}$
Hence, the equation $\frac{\tan x-1}{1-\cot x}$ is equal to $\tan x$.
