Use a sum or difference identity to find the exact value.
tan 25deg + tan 5deg / 1-tan 25deg tan 5deg
tan(x+y) = (tanx + tany)/(1-tanx tany)
so, what you have is
tan(25°+5°) = tan 30° = 1/√3
To find the exact value of the expression tan 25° + tan 5° / 1 - tan 25° tan 5°, we can use the tangent sum identity.
The tangent sum identity states that tan(A + B) = (tan A + tan B) / (1 - tan A tan B).
In this case, let A = 25° and B = 5°. We have:
tan(25° + 5°) = (tan 25° + tan 5°) / (1 - tan 25° tan 5°).
Now, we know that tan(25° + 5°) is actually tan 30° since 25° + 5° = 30°. The tangent of 30° is a known value because it is a 30-60-90 triangle. In a 30-60-90 triangle, the tangent of 30° is √3 / 3.
Therefore, we can rewrite the expression as:
√3 / 3 = (tan 25° + tan 5°) / (1 - tan 25° tan 5°).
Now we can solve for (tan 25° + tan 5°) / (1 - tan 25° tan 5°):
(tan 25° + tan 5°) / (1 - tan 25° tan 5°) = √3 / 3.
Therefore, the exact value of the given expression is √3 / 3.