Question

How do you solve `tan^2theta=1`?

Answer 1

`theta = \pm pi/8`

Explanation:

If you want a certain quantity to equal `1` when squared, then your quantity must be either `1` or `-1`. Every other number would not equal `1` when squared.

Since our quantity is `tan(theta)`, we're asking for `tan(theta)=1` or `tan(theta)=-1`

You could either look for a table of known values to find which angles `theta` satisfy these requests, or you can remember that, by definition,

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

So, `tan(theta)=1` leads to

`sin(theta)/cos(theta)=1 \iff sin(theta)=cos(theta)`

and the sine and cosine functions have the same value where `theta = 45°= pi/4` radians.

For the same reason, `tan(theta)=-1` leads to `sin(theta)=-cos(theta)`, which happens for `theta=-45°=-pi/8` radians. You can find this second solution remembering that the tangent is an odd function, i.e. `tan(-x)=-tan(x)`.