Simplify (sin(theta)cos(theta))/(sin(theta-pi))

Solution:

Step 1:

Decompose the expression into two separate fractions: $\frac{\cos(\theta)}{1} \cdot \frac{\sin(\theta)}{\sin(\theta - \pi)}$.

Step 2:

Transform $\frac{\sin(\theta)}{\sin(\theta - \pi)}$ into a multiplication expression: $\frac{\cos(\theta)}{1} \cdot \left( \sin(\theta) \cdot \frac{1}{\sin(\theta - \pi)} \right)$.

Step 3:

Express $\sin(\theta)$ as a fraction over $1$: $\frac{\cos(\theta)}{1} \cdot \left( \frac{\sin(\theta)}{1} \cdot \frac{1}{\sin(\theta - \pi)} \right)$.

Step 4:

Proceed to simplify the expression.

Step 4.1:

Divide $\sin(\theta)$ by $1$: $\frac{\cos(\theta)}{1} \cdot \left( \sin(\theta) \cdot \frac{1}{\sin(\theta - \pi)} \right)$.

Step 4.2:

Change $\frac{1}{\sin(\theta - \pi)}$ to $\csc(\theta - \pi)$: $\frac{\cos(\theta)}{1} \cdot \left( \sin(\theta) \cdot \csc(\theta - \pi) \right)$.

Step 5:

Divide $\cos(\theta)$ by $1$ to obtain the final simplified result: $\cos(\theta) \cdot \sin(\theta) \cdot \csc(\theta - \pi)$.

Knowledge Notes:

1. Trigonometric Functions: Sine (sin), cosine (cos), and cosecant (csc) are basic trigonometric functions. The cosecant is the reciprocal of the sine function, i.e., $\csc(\theta) = \frac{1}{\sin(\theta)}$.

2. Trigonometric Identities: There are several identities in trigonometry that can simplify expressions, such as the Pythagorean identities, angle sum and difference identities, and reciprocal identities.

3. Simplification of Fractions: When simplifying expressions involving fractions, it's often useful to separate or combine fractions and to express trigonometric functions in their reciprocal forms.

4. Angle Transformation: The angle $\theta - \pi$ represents an angle that is $\pi$ radians less than $\theta$. In the unit circle, this corresponds to a reflection over the x-axis.

5. LaTeX Formatting: Expressions involving trigonometric functions and fractions can be neatly formatted using LaTeX syntax, which improves the readability of mathematical expressions.