Let's start by breaking down and simplifying the given expression step-by-step:
Given:
[tex]\[ \csc(-\beta) \cdot \tan(\beta) \][/tex]
### Step 1: Use the Trigonometric Identity for Cosecant
Recall that \(\csc(x) = \frac{1}{\sin(x)}\). Additionally, the cosecant function has this property when dealing with negative angles:
[tex]\[ \csc(-\beta) = -\csc(\beta) \][/tex]
Thus,
[tex]\[ \csc(-\beta) = -\frac{1}{\sin(\beta)} \][/tex]
### Step 2: Substitution
Substitute \(\csc(-\beta)\) with \(-\frac{1}{\sin(\beta)}\) in the given expression:
[tex]\[ \csc(-\beta) \cdot \tan(\beta) = \left( -\frac{1}{\sin(\beta)} \right) \cdot \tan(\beta) \][/tex]
### Step 3: Express Tangent in Terms of Sine and Cosine
Recall that \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). So we can rewrite \(\tan(\beta)\):
[tex]\[ \tan(\beta) = \frac{\sin(\beta)}{\cos(\beta)} \][/tex]
### Step 4: Substitute Tangent Expression
Now, substitute \(\tan(\beta)\) in the simplified expression:
[tex]\[ -\frac{1}{\sin(\beta)} \cdot \frac{\sin(\beta)}{\cos(\beta)} \][/tex]
### Step 5: Simplify the Expression
When we multiply these terms:
[tex]\[ -\frac{1}{\sin(\beta)} \cdot \frac{\sin(\beta)}{\cos(\beta)} = -\frac{\sin(\beta)}{\sin(\beta) \cos(\beta)} = -\frac{1}{\cos(\beta)} \][/tex]
Thus, the simplified expression is:
[tex]\[ -\frac{1}{\cos(\beta)} \][/tex]
### Step 6: Use the Trigonometric Identity for Secant
Recall that \(\sec(x) = \frac{1}{\cos(x)}\). This allows another way to express the result:
[tex]\[ -\frac{1}{\cos(\beta)} = -\sec(\beta) \][/tex]
However, keeping it in the fraction form is often more useful for clarity in various contexts.
### Final Answer
After simplifying the given expression \( \csc(-\beta) \cdot \tan(\beta) \), we get:
[tex]\[ -\frac{1}{\cos(\beta)} \][/tex]
