sin squared theta minus cos squared theta divided by sin theta times cos theta

To simplify the expression \(\frac{\sin^2 \theta - \cos^2 \theta}{\sin \theta \cos \theta}\), let's proceed with the following steps:

### Step 1: Recognize an Identity

The numerator \(\sin^2 \theta - \cos^2 \theta\) resembles a known trigonometric identity. Recall the identity:

\[
\cos 2\theta = \cos^2 \theta - \sin^2 \theta
\]

However, \(\sin^2 \theta - \cos^2 \theta\) is the negative of \(\cos 2\theta\):

\[
\sin^2 \theta - \cos^2 \theta = -(\cos^2 \theta - \sin^2 \theta) = -\cos 2\theta
\]

### Step 2: Substitute in the Simplified Expression

Substituting \(-\cos 2\theta\) in the numerator, the expression becomes:

\[
\frac{-\cos 2\theta}{\sin \theta \cos \theta}
\]

### Step 3: Use the Double Angle Formula for Sine in the Denominator

Recall that the double angle formula for sine is:

\[
\sin 2\theta = 2 \sin \theta \cos \theta
\]

Thus, \(\sin \theta \cos \theta = \frac{\sin 2\theta}{2}\).

### Step 4: Simplify the Expression

Replacing the denominator with \(\frac{\sin 2\theta}{2}\), the expression now is:

\[
\frac{-\cos 2\theta}{\frac{\sin 2\theta}{2}} = -\frac{\cos 2\theta \cdot 2}{\sin 2\theta} = -2 \frac{\cos 2\theta}{\sin 2\theta}
\]

Since \(\frac{\cos 2\theta}{\sin 2\theta} = \cot 2\theta\), the simplified expression is:

\[
-2 \cot 2\theta
\]

So, the simplified form of \(\frac{\sin^2 \theta - \cos^2 \theta}{\sin \theta \cos \theta}\) is:

\[
-2 \cot 2\theta
\]

Would you like more details or have any questions?

### Related Questions:

1. How do you simplify expressions involving other trigonometric identities?
2. What are some common applications of the double angle formulas?
3. How do you convert trigonometric expressions into terms of a single trigonometric function?
4. Can you derive the double angle formulas from the addition formulas?
5. How does the cotangent function relate to other trigonometric functions?

**Tip:** When simplifying trigonometric expressions, always look for identities or patterns, such as double angle or Pythagorean identities, that can simplify your work.

Discover how to simplify the trigonometric expression (sin^2 θ - cos^2 θ) / (sin θ cos θ) using double angle identities. This step-by-step solution explains key concepts such as double angle formulas and cotangent function, perfect for students in Grades 10-12.

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