Math
posted by Anonymous on .
Solve for x:
sec(2x)csc(2x) = 2csc(x)
for 0 < x < 2pi
[the first "<" sign is less than or equal to]
Thank you!

sec(2x)csc(2x) = 2csc(x)
1/((cos 2x)(sin 2x)) = 2/sinx
crossmultiply
2(cos 2x)(sin 2x) = sinx
2(cos 2x)(2sinxcosx) = sinx
divide by sinx
2(cos 2x)(2cosx) = 1
4cosx(2cos^2 x  1) = 1
8cos^3 x  4cosx  1 = 0
let's let cosx = a
so we are solving
8a^3  4a  1 = 0
after a few tries I go a = 1/2 to work
giving me
(2a1)(4a^2  2a  1) = 0
a = .5 or a = .809 or .309
so cosx = .5 or cosx = .809 or .309
I will do one of these:
cosx = .5, the reference angle is pi/3 or 60º
but the cosine is negative in quadrants II and III
so x = pi  pi/3 = 2pi/3 or 18060 = 120º
or x = pi + pi/3 = 4pi/3 or 180+60 = 240º
You should have 6 different answers 
Left hand side 
sec(2x)csc(2x)= 1/ cos(2x) 1/sin(2x)
= 1/(Cos2x)* 1/(2 Sinx Cosx)
Right Hand side = 1/2Sinx
Equating LHS & RHS
1/{Cos2x}* 1/(2 Sinx Cosx)
= 1/2Sinx
or
1/{Cos2x}* 1/cos x = 1
or {Cos2x}* Cosx = 1
this actually only holds true for x = 0 and x = 2pi 
Vipster, if x = 0, the original equation has undefined calculations, csc 0 is undefined.
you have an error by saying
2csc(x) = 1/2Sinx
2csc(x) = 2/sinx