SOLUTION: If tan3A/tanA = k, then find cos3A/cosA

The numerator and denominator are terms of the double angle formulas:



Since there are two formulas we could add and subtract the second
terms to make two terms on top and bottom if we first multiplied
top and bottom by 2:



Now break the multiplication by 2 into the addition of two of them:



Reverse the order of the factors in the denominator to make it jibe
with the first terms of the two double-angle formulas:



Add and subtract the second terms of the double angle formulas:







Use the identity  in denominator



Use the identity  on first terms of numerator and denominator:



Factor numerator and denominator:



Cancel sin(2A)'s





(1)     

Now let



We want to find p.  I tried simplifying ,
but that didn't pan out to be useful.  So I tried simplifying k∙p,
instead, and it worked. 



Cancel:



Write 3A as 2A+A and A as 2A-A













Cancel:



To find p, we simply divide k∙p by k, using (1) above:





Cancel and we have:



Or if you prefer another form:







Edwin