How can I find the exact value of tan(7π/8) using the half-angle identity?

To find the exact value of tan(7π/8), we can use the half-angle identity for tangent. The half-angle identity states that:

tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))

In our case, we can express 7π/8 as 7π/8 = π - π/8. This allows us to rewrite the tangent function:

tan(7π/8) = tan(π - π/8) = -tan(π/8)

Since tan(π - x) = -tan(x), we can then focus on calculating tan(π/8). We can use the half-angle identity by letting θ = π/4 (which is a known angle with an easily calculated tangent):

tan(π/8) = tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))

First, we need to find the cosine of π/4:

cos(π/4) = √2 / 2

Putting this value into the half-angle identity:

tan(π/8) = √((1 - √2/2) / (1 + √2/2))

Now, simplifying further:

1 - √2/2 = (2 - √2) / 2

1 + √2/2 = (2 + √2) / 2

Now substituting back:

tan(π/8) = √(((2 - √2) / 2) / ((2 + √2) / 2)) = √((2 - √2) / (2 + √2))

This gives:

tan(π/8) = √((2 - √2) / (2 + √2))

Now let’s look at the overall value for tan(7π/8):

tan(7π/8) = -tan(π/8) = -√((2 - √2) / (2 + √2))

So, the exact value of tan(7π/8) is:

tan(7π/8) = -√((2 - √2) / (2 + √2))

This is the simplified expression for the tangent of 7π/8.