solve the equation on the interval 0,2π).\ntan x(cot x + 1)=0\nselect the correct choice below and, if…

solve the equation on the interval 0,2π).\ntan x(cot x + 1)=0\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. x = (type an exact answer, using π as needed. use a comma to separate answers as needed. type your answer in radians. use integers or fractions for any numbers in the expression.)\n0,\\frac{3\\pi}{7\\pi},\\frac{\\pi}{4},\\frac{\\pi}{4}\nb. there is no solution.

solve the equation on the interval 0,2π).\ntan x(cot x + 1)=0\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. x = (type an exact answer, using π as needed. use a comma to separate answers as needed. type your answer in radians. use integers or fractions for any numbers in the expression.)\n0,\\frac{3\\pi}{7\\pi},\\frac{\\pi}{4},\\frac{\\pi}{4}\nb. there is no solution.

Answer

Explanation:

Step1: Solve (\tan x = 0)

We know that (\tan x=\frac{\sin x}{\cos x}). When (\tan x = 0), (\sin x=0) and (\cos x\neq0). On the interval ([0,2\pi)), (\sin x = 0) when (x = 0,\pi).

Step2: Solve (\cot x+1 = 0)

Since (\cot x=\frac{\cos x}{\sin x}), the equation (\cot x+1 = 0) can be rewritten as (\frac{\cos x}{\sin x}+1 = 0), or (\frac{\cos x+\sin x}{\sin x}=0). This implies (\cos x+\sin x = 0) ((\sin x\neq0)). Dividing both sides by (\cos x) (assuming (\cos x\neq0)), we get (1+\tan x = 0), so (\tan x=- 1). On the interval ([0,2\pi)), (\tan x=-1) when (x=\frac{3\pi}{4},\frac{7\pi}{4}).

Answer:

(x = 0,\frac{3\pi}{7},\pi,\frac{7\pi}{4})