To solve the problem of finding the value of [tex]\(\frac{\tan 80^\circ}{\cot 10^\circ}\)[/tex], let's start by exploring the trigonometric identities and relationships involved.
### Step-by-Step Solution:
1. Recall the Definition of Cotangent:
Cotangent is the reciprocal of tangent. Thus, [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex]. Given this, we can express [tex]\(\cot 10^\circ\)[/tex] as:
[tex]\[
\cot 10^\circ = \frac{1}{\tan 10^\circ}
\][/tex]
2. Substitute into the Expression:
Using the fact that [tex]\(\cot 10^\circ = \frac{1}{\tan 10^\circ}\)[/tex], we substitute it into the given expression:
[tex]\[
\frac{\tan 80^\circ}{\cot 10^\circ} = \frac{\tan 80^\circ}{\frac{1}{\tan 10^\circ}} = \tan 80^\circ \times \tan 10^\circ
\][/tex]
3. Use the Complementary Angle Relationship:
One of the important trigonometric identities is that [tex]\(\tan(90^\circ - \theta) = \cot \theta\)[/tex]. Applying this, we get:
[tex]\[
\tan 80^\circ = \cot (90^\circ - 80^\circ) = \cot 10^\circ
\][/tex]
4. Substitute into the Simplified Expression:
Now substitute [tex]\(\cot 10^\circ\)[/tex] back into the simplified form:
[tex]\[
\tan 80^\circ = \cot 10^\circ
\][/tex]
5. Calculate the Ratio:
Given that [tex]\(\tan 80^\circ = \cot 10^\circ\)[/tex], we can now write:
[tex]\[
\frac{\tan 80^\circ}{\cot 10^\circ} = \frac{\cot 10^\circ}{\cot 10^\circ} = 1
\][/tex]
Thus, the value of [tex]\(\frac{\tan 80^\circ}{\cot 10^\circ}\)[/tex] is:
[tex]\[
\boxed{1}
\][/tex]
This complete analytical approach confirms that the ratio indeed equals 1 without the need for any numerical approximation.
