Question

How do you solve `secxcscx - 2cscx = 0`?

Answer 1

Factorize the left hand side and equate the factors to zero.
Then, use the notion that : `secx=1/cosx" "` and `cscx=1/sinx`

Result : `color(blue)(x=+-pi/3+2pi"k , k"in ZZ )`

Explanation:

Factorizing takes you from
`secxcscx-2cscx=0`
to
`cscx(secx-2)=0`

Next, equate them to zero
`cscx=0=> 1/sinx=0`

However, there is no real value of x for which `1/sinx=0`

We move on to `secx-2=0`

`=>secx=2`

`=>cosx=1/2=cos(pi/3)`

`=>x=pi/3`

But `pi/3` is not the only real solution so we need a general solution for all the solutions.

Which is : `color(blue)(x=+-pi/3+2pi"k , k "in ZZ )`

Reasons for this formula:
We include `-pi/3` because `cos(-pi/3)=cos(pi/3)`

And we add `2pi` because `cosx` is of period `2pi`

The General solution for any `"cosine"` function is :

`x=+-alpha+2pi"k , k" in ZZ`

where `alpha` is the principal angle which just an acute angle

For example : `cosx=1=cos(pi/2)`

So `pi/2` is the principal angle!