Trevor Pythagoras

We can use expressions for cos(A+B) and sin(A+B) to help us find tan(A+B).

Using the identity tanx = sinx / cosx we can write

we can now substitue in
sin(A+B) = sinAcosB + sinBcosA
and
cos(A+B) = cosAcosB – sinAsinB
to get

We can now divide both the top and bottom by cosAcosB to get

or

We can now simplify this by cancelling any cosA and cosB to get

finally by substituting the identity we find our result

And it can be shown that this result can be extended to