Question

Question #ce4ab

Answer 1

One cannot prove this as an identity, because it is only true for these values of `theta`:

`theta = pi/2 + npi` and `theta = pi/4 + npi`; `n in ZZ`

Explanation:

If you want the values for `theta`, then starting with:

`cot(theta)cos(theta)csc(theta)= cot(theta)`

Use the identity `csc(theta) = 1/sin(theta)`:

`cot(theta)cos(theta)/sin(theta)= cot(theta)`

Use the identity `cos(theta)/sin(theta)= cot(theta)`

`cot^2(theta) = cot(theta)`

Subtract `cot(theta)` from both sides:

`cot^2(theta) - cot(theta)= 0`

Factor:

`cot(theta)(cot(theta) - 1)= 0`

There are two roots:

`cot(theta) = 0 and cot(theta) = 1`

Use the inverse cotangent on both:

`theta = cot^-1(0) and theta = cot^-1(1)`

The primary values for these are:

`theta = pi/2` and `theta = pi/4`

The cotangent function will repeat the roots at integer multiples of `pi`:

`theta = pi/2 + npi` and `theta = pi/4 + npi`; `n in ZZ`