Use trigonometric identities to transform the left side of the equation into the right side (0 < 𝜃 < 𝜋/2).
tan 𝛽 + cot 𝛽
tan 𝛽
 = 	
tan 𝛽
tan 𝛽
 + 
cot 𝛽
tan 𝛽
 

We are given the equation:

\[
\frac{\tan \beta + \cot \beta}{\tan \beta} = \frac{\tan \beta}{\tan \beta} + \frac{\cot \beta}{\tan \beta}
\]

To transform the left-hand side into the right-hand side, let's break it down step by step using trigonometric identities.

### Step 1: Rewrite the left-hand side
The left-hand side is:

\[
\frac{\tan \beta + \cot \beta}{\tan \beta}
\]

Recall that \(\cot \beta = \frac{1}{\tan \beta}\). Substituting this in:

\[
\frac{\tan \beta + \frac{1}{\tan \beta}}{\tan \beta}
\]

### Step 2: Combine terms in the numerator
To combine the terms in the numerator, let's rewrite them as a single fraction:

\[
\frac{\tan^2 \beta + 1}{\tan \beta}
\]

So the left-hand side becomes:

\[
\frac{\frac{\tan^2 \beta + 1}{\tan \beta}}{\tan \beta} = \frac{\tan^2 \beta + 1}{\tan^2 \beta}
\]

### Step 3: Simplify the right-hand side
Now, let's simplify the right-hand side, which is:

\[
\frac{\tan \beta}{\tan \beta} + \frac{\cot \beta}{\tan \beta}
\]

The first term simplifies to 1:

\[
1 + \frac{\cot \beta}{\tan \beta}
\]

Since \(\cot \beta = \frac{1}{\tan \beta}\), the second term becomes:

\[
1 + \frac{\frac{1}{\tan \beta}}{\tan \beta} = 1 + \frac{1}{\tan^2 \beta}
\]

### Step 4: Compare the expressions
Now we have:

- Left-hand side: \(\frac{\tan^2 \beta + 1}{\tan^2 \beta}\)
- Right-hand side: \(1 + \frac{1}{\tan^2 \beta}\)

Both expressions are identical, so the transformation is complete.

Thus, the equation holds:

\[
\frac{\tan \beta + \cot \beta}{\tan \beta} = \frac{\tan \beta}{\tan \beta} + \frac{\cot \beta}{\tan \beta}
\]

Would you like more details or have any questions? Here are some related questions to deepen your understanding:

1. How do trigonometric identities like \(\cot \beta = \frac{1}{\tan \beta}\) help in simplifying equations?
2. What is the significance of the interval \(0 < \beta < \frac{\pi}{2}\) for trigonometric functions?
3. Can this equation be transformed using a different set of trigonometric identities?
4. How would the equation change if you replaced \(\tan \beta\) with \(\sin \beta / \cos \beta\)?
5. What other identities could be useful in simplifying similar expressions?

**Tip**: Familiarizing yourself with the fundamental trigonometric identities can help simplify a wide range of complex trigonometric expressions.

Use trigonometric identities to transform the left side of the equation into the right side (0 < 𝜃 < 𝜋/2). tan 𝛽 + cot 𝛽 / tan 𝛽 = tan 𝛽 / tan 𝛽 + cot 𝛽 / tan 𝛽

Explore how to simplify the trigonometric expression (tan 𝛽 + cot 𝛽) / tan 𝛽 using identities like cot 𝛽 = 1 / tan 𝛽. This step-by-step solution is ideal for students in Grades 10-12 and covers key trigonometric concepts.

Math Problem Statement

Use trigonometric identities to transform the left side of the equation into the right side (0 < 𝜃 < 𝜋/2). tan 𝛽 + cot 𝛽 tan 𝛽

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