Identities of cos(x) and sin(x)
Find x if :
1/sin(x)-1/cos(x)=2*sqrt(2)
Solution :
Squared for both sides :
[1/sin(x)-1/cos(x)]^2=8
[cos(x)-sin(x)]^2/[sin(x)*cos(x)]^2=8
[cos^2(x)-2*cos(x)*sin(x)+sin^2(x)]/[2*sin(x)*cos(x)/2]^2=8
[1-sin(2x)]/[sin^2(2x)/4]=8
4*[1-sin(2x)]/sin^2(2x)=8
divide both sides by 4 :
[1-sin(2x)]/sin^2(2x)=2
multiply both sides by sin^2(2x) :
1-sin(2x)=2*sin^2(2x)
2*sin^2(2x)+sin(2x)-1=0
2*sin^2(2x)+2*sin(2x)-sin(2x)-1=0
2*sin(2x)*[sin(2x)+1]-[sin(2x)+1]=0
[sin(2x)+1]*[2*sin(2x)-1]=0
sin(2x)+1=0
sin(2x)=-1
2x=270+k*360
if k=0 then 2x=270
x1=135 deg
if k=1 then 2x=630
x2=315deg
2sin(2x)-1=0
sin(2x)=1/2
2x=30+k*360
k=0 then 2x=30
x3=15deg
k=1 then 2x=390
x4=195 deg
2x=150+k*360
k=0 then 2x=150
x5=75
k=1 then 2x=510
x6=255deg
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