SOLUTION: show that 1+cos(x)/sin(x)=sin(x)/1-cos(x)

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Question 282124: show that 1+cos(x)/sin(x)=sin(x)/1-cos(x)
Answer by dabanfield(803) About Me  (Show Source):
You can put this solution on YOUR website!
show that 1+cos(x)/sin(x)=sin(x)/1-cos(x)
Let's start with the rigth-hand side above:
sin(x)/(1-cos(x))
Multiply above numerator and denominator by 1+cos(x):
(sin(x)*(1+cos(x))/((1-cos(x))*(1+cos(x))
(sin(x)+sin(x)*cos(x))/(1^2 -cos(x)+cos(x)-cos(x)^2)
(sin(x)+sin(x)*cos(x))/(1-cos(x)^2)
Using the identity sin(x)^2 + cos(x)^2 = 1 we know sin(x)^2 = 1 - cos(x)^2. So the above becomes:
(sin(x)+sin(x)*cos(x))/(sin(x)^2 =
(sin(x)/sin(x)^2) + (sin(x)*cos(x)/sin(x)^2) =
1/sin(x) + cos(x)/sin(x) =
(1 + cos(x))/sin(x)