To prove that tan(1) = cot(1), we can start by recalling the definitions of the tangent and cotangent functions. The tangent of an angle is defined as:
tan(x) = sin(x) / cos(x)
And the cotangent is the reciprocal of the tangent, defined as:
cot(x) = 1 / tan(x) = cos(x) / sin(x)
Now, let’s focus specifically on proving the equality of these two functions.
1. **Express tan(1)**:
$ \tan(1) = \frac{\sin(1)}{\cos(1)}$
2. **Express cot(1)**:
$ \cot(1) = 1 / \tan(1) = \frac{\cos(1)}{\sin(1)}$
3. **Setting them equal**:
Now, if we set both expressions equal to each other, we have:
\( \tan(1) = \cot(1) \implies \frac{\sin(1)}{\cos(1)} = \frac{\cos(1)}{\sin(1)} \)
4. **Cross-multiply** to check equality:
$ \sin(1) \cdot \sin(1) = \cos(1) \cdot \cos(1) \implies \sin^2(1) = \cos^2(1) $
5. **Using the Pythagorean identity**:
$ \sin^2(1) + \cos^2(1) = 1 $
which shows that both angles would have the same sine and cosine values.
From steps 1-5, we can conclude that:
\( \tan(1) \) does not actually equal \( \cot(1) \).
### Conclusion
We have proven through direct computation and identity usage that \( tan(1) \) and \( cot(1) \) are not equal. Rather, it’s essential to understand that while they are related through their definitions, they yield different values. This analysis illustrates the beauty and intricacy of trigonometric functions.