SOLUTION: Prove cot(45�-A) = {{{ (cotA + 1)/ (cotA - 1) }}} Hence show that cot15� = 2 + sqrt3

Question 217335: Prove cot(45�-A) =
Hence show that cot15� = 2 + sqrt3
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
cot(45�-A) =

There may be more clever ways of doing this but one approach to a lot of Trig. problems is to start by rewriting the expression in terms of sin and cos. Since cot(a) = cos(a)/sin(a):
cot(45�-A) =
Now we can use the angle difference identitites: and to rewrite the right side:
cot(45�-A) =
Since :
cot(45�-A) =
We can reduce the fraction by factoring out :
cot(45�-A) =
Now the 's cancel leaving:
cot(45�-A) =
Looking at what we have and at where we want to be we can see that we're very close. All we need is sin(A) as a denominator of each term. So we multiply the top and bottom by :
cot(45�-A) =
Using the Distributive Property we get:
cot(45�-A) =
which simplifies to
cot(45�-A) =

To find cot(15�): If we realize that 15 = 45-30 then we can use the formula above by setting A = 30:
cot(15�) = cot(45�-30) =

Substituting this we get:
cot(15�) = cot(45�-30) =
Next we rationalize the denominator. To do this we will use the conjugate of the denominator which is :
cot(15�) = cot(45�-30) =
Multiplying this out (using either FOIL or the patterns for (a+b)(a+b) and (a+b)(a-b)) we get:
cot(15�) = cot(45�-30) =
cot(15�) = cot(45�-30) =
See how the conjugate made the denominator rational?
cot(15�) = cot(45�-30) =
Reduce the fraction by factoring out 2 and canceling:
cot(15�) = cot(45�-30) =
cot(15�) = cot(45�-30) =

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