Certainly! Let's transform the left side of the given equation step by step using trigonometric identities.
We start with the given equation:
[tex]\[
\frac{\tan \beta + \cot \beta}{\tan \beta}
\][/tex]
### Step 1: Separate the terms
We can split the numerator into two separate fractions:
[tex]\[
\frac{\tan \beta}{\tan \beta} + \frac{\cot \beta}{\tan \beta}
\][/tex]
### Step 2: Simplify each term
Next, we simplify each fraction individually. For the first term:
[tex]\[
\frac{\tan \beta}{\tan \beta} = 1
\][/tex]
For the second term, note that cotangent [tex]\( \cot \beta \)[/tex] is the reciprocal of tangent [tex]\( \tan \beta \)[/tex]:
[tex]\[
\frac{\cot \beta}{\tan \beta} = \frac{1/\tan \beta}{\tan \beta} = \frac{1}{\tan^2 \beta}
\][/tex]
So, combining these results, we have:
[tex]\[
1 + \frac{1}{\tan^2 \beta}
\][/tex]
### Step 3: Use trigonometric identities
We recognize that the term [tex]\( \frac{1}{\tan^2 \beta} \)[/tex] can be rewritten using a known trigonometric identity involving the cosecant function. The identity states:
[tex]\[
1 + \cot^2 \beta = \csc^2 \beta
\][/tex]
Since [tex]\( \cot \beta = \frac{1}{\tan \beta} \)[/tex], we have:
[tex]\[
\cot^2 \beta = \left(\frac{1}{\tan \beta}\right)^2 = \frac{1}{\tan^2 \beta}
\][/tex]
Thus, our expression:
[tex]\[
1 + \frac{1}{\tan^2 \beta} = 1 + \cot^2 \beta = \csc^2 \beta
\][/tex]
### Conclusion:
By applying the trigonometric identity, we have:
[tex]\[
\frac{\tan \beta + \cot \beta}{\tan \beta} = 1 + \frac{1}{\tan^2 \beta} = \csc^2 \beta
\][/tex]
Therefore, the completely simplified form of the expression on the left side is:
[tex]\[
\csc^2 \beta
\][/tex]
