solve for the TWO values of x that satisfy the equation `(1/16)csc^2(x/4)-cos^2(x/4)=0`    on  0° < x < 360°     express exact solutions in degrees.  

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tiburtius eNotes educator| Certified Educator


Since by definition cosecant is `csc z=1/sin z` we have




Since `2sin z cos z=sin(2z)` we have


Now we take quare root and get two solutions

1) `2sin(x/2)=1`

2) `2sin(x/2)=-1`

1) `sin(x/2)=1/2`

`x/2=-30^o +k360^o`

`x=-60^o + k720^o`    <-- First solution

`x/2=210^o + k360^o`

`x=420^o +k720^o` <-- Second solution

2) `sin(x/2)=-1/2`

`x/2=30^o +k360^o`

`x=60^o + k720^o" <--Third solution"`

`x/2=150^o +k360^o`

`x=300^o +k720^o" <-- Fourth solution"`

Now since first and second solution don't lie between 0° and 360° you have only two solutions and they are 60° and 300°.