Challenge Trigonometry Explanation, i don't understand what the question is asking me for.

Question

So my teacher asked us challenge problems for this

1.Is the statement tan(x-y)=tan(x)-tan(y) true for all values of x and y?

2.Given the sum and difference identities for sine and cosine, explain how to derive an identity for tan(?-β)?

3.Show how you can use your result to derive an identity for tan(π-θ)?

Answer 1

1. The statement is false. We only need on counter-example to prove it is false. Here is an example:

 

  tan(89.999^o - 45^o) = tan(44.999^o)=1

 

  but tan(89.999^o) - tan(45^o) = 57295 - 1 = 57295, apparently they are not equal. When the angle is very close to 90⁰,

 

  tangent function will be close to infinity.

 

2. Since you need to derive tan(α - β), you only need the difference identities for sine and cosine. Here is the procedure:

 

Given     Sin(α - β) = sinαcosβ - cosαsinβ

  

        and cos(a-β) =  cosαcosβ + sinαsinβ

 

then  tan(α - β) =   sin(α-β)

                            ----------  

                            cos(α -β)

 

                       

                        sinαcosβ - cosαsinβ

                    = ----------------------           

                         cosαcosβ + sinαsinβ

 

 

Now divide both sides by cosαcosβ to create tanα and tanβ :

 

                        tanα - tanβ

                   =  --------------

                        1  + tanαtanβ

 

                                                                         tanα - tanβ
 So we get the tangent identity :  tan(α-β) =       --------------
                                                                        1 + tanαtanβ   

 

 

3. To find tan(π - θ). use the tangent identity from 2):

 

                     tanπ - tanθ

  tan(π - θ) =  -------------

                      1 + tanπtanθ

 

                     0 - tanθ

                =  ----------

                     1+ 0tanθ

 

                     -tanθ

                =  ---------

                        1

 

               = - tanθ

 

Let me know if you have any questions.