Question

cosθ/(cos(θ)+sin(θ)-1)-sin(θ)/(cos(θ)+sin(θ)-1)+1/(cos(θ)+sin(θ)-1)=csc(θ)+cot(θ)

Answer 1

$$\begin{array}{l} \frac{\cos t}{\cos t + \sin t - 1} - \frac{\sin t}{\cos t + \sin t - 1} + \frac{1}{\cos t + \sin t - 1} = \\ = \frac{\cos t - \sin t + 1}{\cos t + \sin t - 1} = \frac{\cos t - \sin t + 1}{\cos t + \sin t - 1} \cdot \frac{\cos t + \sin t + 1}{\cos t + \sin t + 1} = \\ = \frac{(\cos t + 1)^2 - \sin^2 t}{(\cos t + \sin t)^2 - 1} = \frac{\cos^2 t + (1 - \sin^2 t) + 2 \cos t}{\sin 2 t} = \\ = \frac{2 \cos t + 2 \cos^2 t}{2 \sin t \cos t} = \frac{1}{\sin t} + \frac{\cos t}{\sin t} = \csc t + \cot t \\ \end{array}$$ $$\text{If } \cos t + \sin t + 1 = 0 \text{ then } \sqrt{2} \sin \left(t + \frac{\pi}{4}\right) = -1 \Rightarrow t = 5\pi/4 \text{ or } t = -\pi/2$$ $$\begin{array}{l} \frac{\cos t - \sin t + 1}{\cos t + \sin t - 1} = \frac{-2 \sin t}{-2} = \sin t = [t = -\pi/2] = -1 \\ \csc t + \cot t = \frac{1}{\sin t} + \frac{\cos t}{\sin t} = \frac{1 + \cos t}{\sin t} = -1 \\ \end{array}$$

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