The exact value of tan5pi/12 is

Step-by-step explanation:

This is the sum identity for tangent(x + y), where x and y are angles measured in radians.  The formula for tan(x + y) is  

The first order of business is to find which 2 angles, when expressed in terms of the denominator 12, add up to equal

We will use the first quadrant angles, and only the "important" ones:

When expressed in terms of the denominator of 12, these angles have the equivalent angles, in order from above:

We need to find the 2 whose numerators add up to a 5.  That would be:

Remember that

and  so

angle x is  and angle y is , making our tangent sum:

Filling that into our formula for the sum of tan(x + y):

It just so happens that

 and

so our formula then becomes

which simplifies to

and then bring up the lower fraction and flip it to multiply giving you:

I have the feeling that you need to rationalize that denominator, and if you do that, the final answer will be:

I know I kind of left you hanging at the very end with rationalizing, but there was so much already that went into this problem in such depth, that I didn't want to risk possibly confusing you even more than I may have already done so.  Try and follow the best that you can.

im not sure

step-by-step explanation: