Prove,

for all such that .

We also conjecture and prove,

and,

First we prove the difference formula for tangent, .

* Proof. * By definition of the tangent,

Next, we prove the sum formula for tangent, .

* Proof. * Again, starting with the definition of tangent,

Finally, the sum formula for the cotangent.

* Proof. * Starting with the definition of the cotangent,