Solve for x:

Question

sec(2x)csc(2x) = 2csc(x)

for 0 < x < 2pi

[the first "<" sign is less than or equal to]

Thank you!

Answer 1

sec(2x)csc(2x) = 2csc(x)
1/((cos 2x)(sin 2x)) = 2/sinx
cross-multiply
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
(2a-1)(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 180-60 = 120º
or x = pi + pi/3 = 4pi/3 or 180+60 = 240º

You should have 6 different answers

Answer 2

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

Answer 3

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