Explain why tan(x+450°) cannot be simplified using the tangent sum 
formulas but can be simplified by using he sine and cosine formulas?

If we tried to use

tan(a+b) = (tana + tanb)/(1 - tana·tanb)

with a = x and b = 450°, then tanb would be tan450°, which
is not defined.

However sin(a+b) = sina·cosb + cosa·sinb and
        cos(a+b) = cosa·cosb - sina·sinb  

would only involve sin450° and cos450° which are defined
respectively as 1 and 0,

then

sin(x+450°) = sinx·cos450°+cosx·sin450° = sinx·0+cosx·1 = cosx

and

cos(x+450°) = cosx·cos450°-sinx·sin450° = cosx·0-sinx·1 = -sinx

therefore

tan(x+450°) = sin(x+450°)/cos(x+450°) = cosx/(-sinx) =

-cosx/sinx = -cotx.

Edwin
AnlytcPhil@aol.com