How do you find the exact value of `tantheta=cottheta` in the interval `0<=theta<360`?

`tantheta - cot theta = 0`

Rewrite using the identities `tantheta = sintheta/costheta` and `cottheta = costheta/sintheta`.

`sintheta/costheta - costheta/sintheta = 0`

Simplify the fraction

`(sin^2theta - cos^2theta)/(sinthetacostheta)= 0`

Multiply both sides of the equation by `sinthetacostheta` to remove the denominator

`sin^2theta - cos^2theta = 0`

Replace `sin^2theta` with `1-cos^2theta` by rearranging this identity: `sin^2theta+cos^2theta-=1`

`1 - cos^2theta - cos^2theta = 0`

`1 - 2cos^2theta = 0`

`-2cos^2theta= -1`

`cos^2theta = 1/2`

`costheta = +-1/sqrt(2)`

`theta=cos^-1(+-1/sqrt2)=45, 135`

These values are the principle values of `theta`, but we need to find all the values of `theta` that satisfy this equation within our given range.

To find the complementary angles, where `y` is `+"ve"`, within one period of the cosinusoidal wave, we need to subtract `theta` from `360`: `a=360-45=315`

To find the complementary angles, where `y` is `-"ve"`, within one period of the cosinusoidal wave, we need to add `theta` to `90`: `b=90+135=225`

So all the values of `theta` within the given range are:

`theta= 45, 135, 225 " and " 315`

Another way to find the other values of theta is this relation:

`theta_n=theta+n(270), costheta>=0`

`theta_n=theta+n(90), costheta<0`

Note that `n` must be an integer. Note that we use `n(360)` only when dealing with the sine and cosine functions; when dealing with the tangent, we use the period `n(180)`.