SOLUTION: Solve {{{(sin(x/2))^2-(sin(x))^2=0}}}

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Question 138824: Solve %28sin%28x%2F2%29%29%5E2-%28sin%28x%29%29%5E2=0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Start with the given equation



Let . So this means that



Replace with u and with 2u


Rewrite as


Replace with





Square



Factor out



Now use the zero product property:


...or...


Now let's solve :





Take the square root of both sides


...or... Take the arcsine of both sides


Since , this means


...or...



However, since is not in the interval [0,2), it is not a solution.


So the first solution is



--------------------------------------------------------------------

Now let's solve :





Subtract 1 from both sides


Divide both sides by -4


Take the square root of both sides:
...or...


So let's solve the first part :





...or...


...or...


However since our interval is positive, the negative answer is not in the interval.


So another part of the solution is



---------------------


Now let's solve the second part :






...or...


...or...



So another part of the solution is




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Answer:


So all together our solutions are: