Question

Question #cf5ae

Answer 1

I tried this:

Explanation:

Ok, it should be:
`2cos^2(x)-1=0`
rearrange:
`cos^2(x)=1/2`
`cos(x)=+-1/sqrt(2)=+-sqrt(2)/2`
(where at the end I rationalized).

Now we need to find suitable values for `x` so that the cosine will be equal to `+-sqrt(2)/2`:
Have a look:

At least in the interval `[0,2pi]` we will get 4 possibilities:
A) `x=pi/4`
B) `x=3/4pi`
C) `x=5/4pi`
D) `x=7/4pi`

this will be repeated periodically every `2pi`

Answer 2

`" The Soln. Set="{2kpi+-pi/4 : k in ZZ}uu{2kpi+-3pi/4 : k in ZZ}.`

Explanation:

`2cos^2x-1=0`

`rArr 2cos^2x=1, or, cos^2x=1/2.`

`rArr cosx=+-1/sqrt2.`

Knowing that, `costheta=cosalpha rArr theta=2kpi+-alpha, k in ZZ.`

`cosx=1/sqrt2=cos(pi/4) rArr x=2kpi+-pi/4, k in ZZ.`

`cosx=-1/sqrt2=cos(3pi/4) rArr x=2kpi+-3pi/4, k in ZZ.`

`:." The Soln. Set="{2kpi+-pi/4 : k in ZZ}uu{2kpi+-3pi/4 : k in ZZ}.`

Enjoy Maths.!