Question

How many values are there for the `cot(3x-pi/4)!=-1`? `[-5,5]`

Answer 1

There are infinitely many values of x that will make the cotangent function not equal to -1 within the domain `[-5,5]`.

Explanation:

Given: `cot(3x-pi/4)!=-1, {x in RR|-5 <= x <=5}`

Use the identity `cot(u) = 1/tan(u)` where `u = 3x-pi/4`:

`1/tan(3x-pi/4)!=-1, {x in RR|-5 <= x <=5}`

`tan(3x-pi/4)!=1/-1, {x in RR|-5 <= x <=5}`

`tan(3x-pi/4)!=-1, {x in RR|-5 <= x <=5}`

`3x-pi/4!=tan^-1(-1), {x in RR|-5 <= x <=5}`

`3x-pi/4!=(3pi)/4+npi, {x in RR|-5 <= x <=5}, n in ZZ`

`3x != pi+npi, {x in RR|-5 <= x <=5}, n in ZZ`

`x != pi/3+npi/3, {x in RR|-5 <= x <=5}, n in ZZ`

There are infinitely many values of x that make the above inequality true within the domain restrictions.