SOLUTION: please help me solve this hence sinx/(1-cosx)=4tanx for 0 < x < 180

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Question 1075848: please help me solve this
hence sinx/(1-cosx)=4tanx for 0 < x < 180
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
sin%28x%29%2F%281-cos%28x%29%29=4tan%28x%29

We multiply through by 1-cos(x) but if we do we
must rule out the case where the denominator
1-cos(x) would be 0, which would be when cos(x) = 1
which would be x=0°  

sin%28x%29=4tan%28x%29%281-cos%28x%29%5E%22%22%29

We change tan(x) to sin(x)/cos(x)

sin%28x%29=4%28sin%28x%29%2Fcos%28x%29%29%281-cos%28x%29%5E%22%22%29

Distributing on the right gives:

sin%28x%29=4sin%28x%29%2Fcos%28x%29-4sin%28x%29%5E%22%22%29

We multiply through by cos(x) but if we do we
must rule out the case where the denominator
cos(x) would be 0, which would be when 
x = 90° or 270°  

sin%28x%29cos%28x%29=4sin%28x%29-4sin%28x%29cos%28x%29

We get 0 on the right:

5sin%28x%29cos%28x%29-4sin%28x%29=0

Factor out sin(x)

sin%28x%29%285cos%28x%29-4%5E%22%22%29=0 

Setting the first factor = 0,

sin%28x%29=0

x = 0° or 180° but we have ruled out x=0°,
so the only answer from setting the 
first factor = 0 is 180°

Setting the second factor = 0

5cos%28x%29-4=0

5cos%28x%29=4

cos%28x%29=4%2F5

Use calculator to get QI answer and
subtract it from 360° to get QIV answer. 
x = 36.86989765° and 323.1301024°

Solutions: 180°, 36.86989765°, 323.1301024°

Edwin