Question 22305
<pre>Explain why tan(x+450°) cannot be simplified using the tangent sum 
formulas but can be simplified by using he sine and cosine formulas?
<b><font size = 3>
If we tried to use

tan(<font face = "symbol">a</font>+<font face = "symbol">b</font>) = (tan<font face = "symbol">a</font> + tan<font face = "symbol">b</font>)/(1 - tan<font face = "symbol">a</font>·tan<font face = "symbol">b</font>)

with <font face = "symbol">a</font> = x and <font face = "symbol">b</font> = 450°, then tan<font face = "symbol">b</font> would be tan450°, which
is not defined.

However sin(<font face = "symbol">a</font>+<font face = "symbol">b</font>) = sin<font face = "symbol">a</font>·cos<font face = "symbol">b</font> + cos<font face = "symbol">a</font>·sin<font face = "symbol">b</font> and
        cos(<font face = "symbol">a</font>+<font face = "symbol">b</font>) = cos<font face = "symbol">a</font>·cos<font face = "symbol">b</font> - sin<font face = "symbol">a</font>·sin<font face = "symbol">b</font>  

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</pre>