SOLUTION: Solve {{{(sin(x/2))^2-(sin(x))^2=0}}}
Algebra.Com
Question 138824: Solve
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
-\sin^2\left(\theta\right)=0 )
Start with the given equation
Let 
. So this means that 
-\sin^2\left(2u\right)=0 )
Replace 
with u and 
with 2u
-\left(\sin\left(2u\right)\right)^2=0 )
Rewrite )
as \right)^2)
-\left(2\sin\left(u\right)\cos\left(u\right)\right)^2=0 )
Replace )
with \cos\left(u\right))
-4\sin^2\left(u\right)\cos^2\left(u\right)=0 )
Square
\left(1-4\cos^2\left(u\right)\right)=0 )
Factor out )
Now use the zero product property:
=0)
...or... =0 )
Now let's solve :
=0 )
=0 )
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 =0 )
:
=-1 )
Subtract 1 from both sides
=\frac{1}{4} )
Divide both sides by -4
Take the square root of both sides:
=\frac{1}{2})
...or... =-\frac{1}{2} )
So let's solve the first part =\frac{1}{2} )
:
...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
===============================================
Answer:
So all together our solutions are:
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