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

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.