Answer
$\theta = 180^{\circ}+360^{\circ}~n$, where $n$ is a non-negative integer.
Work Step by Step
$tan~\theta-sec~\theta = 1$
$\frac{sin~\theta}{cos~\theta}-\frac{1}{cos~\theta} = 1$
$sin~\theta-1 = cos~\theta$
$sin^2~\theta-2~sin~\theta +1 = cos^2~\theta$
$sin^2~\theta-2~sin~\theta +1 = 1-sin^2~\theta$
$2~sin^2~\theta = 2~sin~\theta$
$sin^2~\theta = sin~\theta$
Therefore, $sin~\theta = 0$ or $sin~\theta = 1$
When $sin~\theta = 0$:
$\theta = 0, 180^{\circ}$
When we check these possible solutions in the original equation, only $\theta = 180^{\circ}$ is a valid solution.
When $sin~\theta = 1$:
$\theta = 90^{\circ}$
When we check this possible solution in the original equation, $sec~\theta$ is undefined. Therefore, $\theta = 90^{\circ}$ is not a valid solution.
The only valid solution is $\theta = 180^{\circ}$
In general, the solutions have the form: $\theta = 180^{\circ}+360^{\circ}~n$, where $n$ is a non-negative integer.
