Determine the exact value of the following expression. tan(17π12)tan(17𝜋12)

The given expression is tan(17π/12).

Step 1: Simplify the angle The angle 17π/12 is not a standard angle. We can express it as a sum or difference of standard angles. The standard angles in radians are multiples of π/4 or π/6.

17π/12 can be written as 4π/3 + π/4.

Step 2: Use the formula for tan(a + b) The formula for tan(a + b) is [tan(a) + tan(b)] / [1 - tan(a)tan(b)].

Step 3: Substitute the values Substitute a = 4π/3 and b = π/4 into the formula.

tan(4π/3 + π/4) = [tan(4π/3) + tan(π/4)] / [1 - tan(4π/3)tan(π/4)].

Step 4: Calculate the values tan(4π/3) = tan(π/3) = √3 and tan(π/4) = 1.

Substitute these values into the formula:

[tan(4π/3) + tan(π/4)] / [1 - tan(4π/3)tan(π/4)] = [√3 + 1] / [1 - √3 * 1] = [√3 + 1] / [1 - √3].

Step 5: Simplify the expression Simplify the expression to get the exact value of tan(17π/12).

So, tan(17π/12) = [√3 + 1] / [1 - √3].