Find the exact value of the expression. tan 105 ° \tan105\degree tan 105°

In this problem, we're asked to find the exact value of the expression, the tangent of a 105 degrees. Now, I don't know the tangent of a 105 degrees off the top of my head, but I do know that I can break down this angle in terms of angles that I already know. Now a 105 degrees can be rewritten as 60 degrees plus 45 degrees. So here, I can take the tangent of 60 plus 45 degrees in order to get the exact value that I'm looking for. Now here, I wanna use my tangent formula for the sum. So I can go ahead and expand this out, and I will end up getting the tangent of 60 degrees plus the tangent of 45 degrees and this is divided by 1 minus the tangent of 60 degrees times the tangent of 45 degrees. Now we know all of these trig values from the unit circle, so let's go ahead and plug in the values that we know. Now the tangent of 60 degrees is equal to the square root of 3, and the tangent of 45 degrees is equal to 1. So in my numerator I'm left with the square root of 3 plus 1. Then in my denominator I still have that 1 minus and then again the tangent of 60 degrees is equal to the square root of 3, and the tangent of 45 degrees is equal to 1. But if I'm multiplying the square root of 3 times times 1, that really just leaves me with the square root of 3. So here I have the square root of 3+1 over 1 minuteus the square root of 3. Now this is an answer, but it is not the final simplified answer that we're looking for. Because remember, whenever we have a denominator that has a radical in it, we want to rationalize that denominator. Now in order to do that, we can multiply our denominator by 1 plus the square root of 3, the conjugate. Now remember, if I am multiplying my denominator by that, I also need to multiply my numerator by that in order to not change the value of my expression. Now from here, I can go ahead and multiply everything out. Now in my numerator, I'm gonna go ahead and foil. So here I have my first terms being multiplied and I end up with the square root of 3. Then I have my outside terms, which the square root of 3 times the square of 3 is just going to give me 3. Then I have my inside terms, 1 times 1. And then finally my last terms, 1 times the square root of 3. So that is my numerator fully foiled out. Then in my denominator, since I have a one minus the square root of 3 times 1 plus the square root of 3, I recognize this as a difference of squares. So this is going to give me 1 squared minus the square root of 3 squared. Now let's simplify that. In my numerator, I can go ahead and combine these square roots of 3 to give me 2 times the square root of 3. Then I can combine some more like terms, 3+1, to give me 4. Then in my denominator squaring those values, I end up with 1 minus 3. Now 1 minus 3 is negative 2. So my denominator here ends up being negative 2. But we can do some more simplification because both of these terms are divisible by 2. So going ahead and dividing that, 2 root 3 divided by negative 2 gives me negative root 3. Then 4 divided by negative 2 gives me negative 2. So this is my final answer here, negative square root of 3 minus 2. Thanks for watching and let me know if you have any questions.