The expression tan(9°) * tan(27°) * tan(63°) * tan(81°) can be evaluated using the properties of the tangent function and its symmetrical properties in trigonometry.
First, it’s essential to note that:
- tan(81°) is the same as cot(9°), because tan(90° – x) = cot(x).
- tan(27°) is the same as cot(63°), for the same reason.
This gives us the following simplifications:
- tan(81°) = cot(9°)
- tan(63°) = cot(27°)
Now, using these relationships, we can rewrite the original expression:
tan(9°) * cot(9°) * tan(27°) * cot(27°).
Since tan(x) * cot(x) = 1, we have:
- tan(9°) * cot(9°) = 1
- tan(27°) * cot(27°) = 1
Thus, the entire expression equals:
1 * 1 = 1.
In summary, the value of tan(9°) * tan(27°) * tan(63°) * tan(81°) is 1.