Answer:
We can start by using the tangent formula:
tan(P) = sin(P) / cos(P)
We are not given the values of sin(P) and cos(P) directly, but we can use the Pythagorean identity to relate them:
sin^2(P) + cos^2(P) = 1
We can rearrange this expression to solve for sin(P) in terms of cos(P):
sin^2(P) = 1 - cos^2(P)
sin(P) = ±√(1 - cos^2(P))
Now we can substitute this expression into the tangent formula:
tan(P) = ±√(1 - cos^2(P)) / cos(P)
Substituting the given values and simplifying:
tan(P) = ±√(1 - (2/√85)^2) / (1/√89)
tan(P) = ±√(1 - 4/85) * √89
tan(P) = ±√(81/85) * √89 / 1
We want to express tan P in simplest radical form, so we can simplify the square root by factoring out the greatest perfect square factor of the numerator:
tan(P) = ±(√9/√85) * (√89/1)
tan(P) = ±(3/√85) * (√89/1)
Therefore, the exact value of tan P in simplest radical form is ±(3/√85) * (√89/1).