How can we use the half angle formula to find the exact value of tan 15 degrees?

To find the exact value of tan 15 degrees, we can utilize the half angle formula for tangent. The formula states:

tan(θ/2) = sqrt((1-cos θ)/(1+cos θ))

In our case, we can consider 15 degrees as half of 30 degrees; therefore, we set θ = 30 degrees.

First, we need the cosine of 30 degrees, which is:

cos(30°) = sqrt(3)/2.

Now, substituting this value into the half angle formula, we have:

tan(15°) = tan(30°/2)
          = sqrt((1 - cos(30°)) / (1 + cos(30°)))
          = sqrt((1 - sqrt(3)/2) / (1 + sqrt(3)/2))

Next, let’s simplify the expression inside the square root:

= sqrt((2/2 - sqrt(3)/2) / (2/2 + sqrt(3)/2))
= sqrt((2 - sqrt(3)) / (2 + sqrt(3))).

Now, we rationalize the denominator to make it simpler to compute:

= sqrt((2 - sqrt(3)) / (2 + sqrt(3))) * (2 - sqrt(3)) / (2 - sqrt(3))
= sqrt((2 - sqrt(3))^2 / (1))
= (2 - sqrt(3)).

Thus, the exact value of tan 15 degrees is:

tan 15° = 2 – sqrt(3).