Answer: [tex]\frac{\pi}{2}, \ \frac{5\pi}{4}, \ \frac{3\pi}{2}, \ \frac{7\pi}{4}[/tex]
We can write the four solutions as: pi/2, 5pi/4, 3pi/2, 7pi/4
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Work Shown:
[tex]\sqrt{2}\cot(x)\sin(x)+\cot(x)=0\\\\\cot(x)\left(\sqrt{2}\sin(x)+1\right)=0\\\\\cot(x)=0 \ \text{ or } \ \sqrt{2}\sin(x)+1=0\\\\\frac{\cos(x)}{\sin(x)}=0 \ \text{ or } \ \sqrt{2}\sin(x)=-1\\\\\cos(x)=0*\sin(x) \ \text{ or } \ \sin(x)=\frac{-1}{\sqrt{2}}\\\\\cos(x)=0 \ \text{ or } \ \sin(x)=\frac{-\sqrt{2}}{2}\\\\[/tex]
Use the unit circle or reference sheet to find that [tex]\cos(x)=0[/tex] leads to [tex]x = \frac{\pi}{2} \ \text{ and } \ x = \frac{3\pi}{2}[/tex] when working with the interval [0,2pi)
Also, use of the unit circle would show that [tex]\sin(x)=\frac{-\sqrt{2}}{2}[/tex] leads to [tex]x = \frac{5\pi}{4} \ \text{ and } \ x = \frac{7\pi}{4}[/tex] when working with the interval [0,2pi)
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Overall, the four solutions on the interval [0, 2pi) are
[tex]\frac{\pi}{2}, \ \frac{5\pi}{4}, \ \frac{3\pi}{2}, \ \frac{7\pi}{4}[/tex]
To verify these solutions, plug them in one at a time into the original equation. You should end up with 0 = 0 after simplifying. Make sure your calculator is in radian mode.
