What is the general solution for sin(x)+sin(x)cot(x)=0 ?

The general solution for sin(x)+sin(x)cot(x)=0 is x=(3pi)/4+pin

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sin(x)+sin(x)cot(x)=0

sin (x) + sin (x) cot (x) = 0

x = \frac{3 π}{4} + πn

Solution steps

sin (x) + sin (x) cot (x) = 0

Factor

sin (x) + sin (x) cot (x) : sin (x) (cot (x) + 1)

sin (x) (cot (x) + 1) = 0

Solving each part separately

sin (x) = 0 or cot (x) + 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

cot (x) + 1 = 0 : x = \frac{3 π}{4} + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + πn

Since the equation is undefined for:

2 πn, π + 2 πn

x = \frac{3 π}{4} + πn

arctan (θ) = 1

cos (2 θ) = - \frac{1}{2}, 0 \leq θ \leq 2 π

sin (2 x) = \frac{2}{5}

sin (2 x) + cos (x) = 0, 0 \leq x \leq 2 π

cos (θ) = - 0.5

What is the general solution for sin(x)+sin(x)cot(x)=0 ?
The general solution for sin(x)+sin(x)cot(x)=0 is x=(3pi)/4+pin