solve the equation on the interval $0,2\\pi)$.\n$(\\cot\\theta - 1)(\\sin\\theta - 1)=0$\nselect the correct…

solve the equation on the interval $0,2\\pi)$.\n$(\\cot\\theta - 1)(\\sin\\theta - 1)=0$\nselect the correct choice below and fill in any answer boxes in your choice.\n$\\bigcirc$ a. $\\theta =$\n(simplify your answer. type an exact answer, using $\\pi$ as needed. type your answer in radians. use integers or\nfractions for any numbers in the expression. use a comma to separate answers as needed.)\n$\\bigcirc$ b. there is no solution on this interval.

solve the equation on the interval $0,2\\pi)$.\n$(\\cot\\theta - 1)(\\sin\\theta - 1)=0$\nselect the correct choice below and fill in any answer boxes in your choice.\n$\\bigcirc$ a. $\\theta =$\n(simplify your answer. type an exact answer, using $\\pi$ as needed. type your answer in radians. use integers or\nfractions for any numbers in the expression. use a comma to separate answers as needed.)\n$\\bigcirc$ b. there is no solution on this interval.

Answer

Explanation:

Step1: Use the zero - product property

If ((\cot\theta - 1)(\sin\theta - 1)=0), then either (\cot\theta - 1 = 0) or (\sin\theta - 1=0).

Step2: Solve (\cot\theta - 1 = 0)

(\cot\theta=1), since (\cot\theta=\frac{\cos\theta}{\sin\theta}), and (\cot\theta = 1) implies (\cos\theta=\sin\theta). On the interval ([0,2\pi)), (\theta=\frac{\pi}{4},\frac{5\pi}{4}).

Step3: Solve (\sin\theta - 1 = 0)

(\sin\theta=1). On the interval ([0,2\pi)), (\theta=\frac{\pi}{2}).

Answer:

(\theta=\frac{\pi}{4},\frac{\pi}{2},\frac{5\pi}{4})