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The given equation is:

\[
\cot \theta = \frac{1}{\sqrt{3}}
\]

We are asked to prove:

\[
\frac{\sin^2 \theta - \sec^2 \theta}{\sin^2 \theta + \sec^2 \theta}
\]

### Step 1: Solving for \(\theta\)

From the equation \(\cot \theta = \frac{1}{\sqrt{3}}\), we know:

\[
\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{1}{\sqrt{3}}
\]

So,

\[
\frac{\cos \theta}{\sin \theta} = \frac{1}{\sqrt{3}} \quad \Rightarrow \quad \cos \theta = \frac{\sin \theta}{\sqrt{3}}
\]

Now, squaring both sides:

\[
\cos^2 \theta = \frac{\sin^2 \theta}{3}
\]

### Step 2: Using the Pythagorean Identity

We know the identity \(\sin^2 \theta + \cos^2 \theta = 1\). Substituting \(\cos^2 \theta = \frac{\sin^2 \theta}{3}\) into this identity:

\[
\sin^2 \theta + \frac{\sin^2 \theta}{3} = 1
\]

Multiplying through by 3 to eliminate the fraction:

\[
3\sin^2 \theta + \sin^2 \theta = 3
\]

This simplifies to:

\[
4\sin^2 \theta = 3 \quad \Rightarrow \quad \sin^2 \theta = \frac{3}{4}
\]

Thus,

\[
\sin \theta = \frac{\sqrt{3}}{2}
\]

### Step 3: Finding \(\sec^2 \theta\)

We know that \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\). From \(\sin^2 \theta = \frac{3}{4}\), we can substitute it into \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\cos^2 \theta\):

\[
\cos^2 \theta = 1 - \frac{3}{4} = \frac{1}{4}
\]

Therefore, \(\sec^2 \theta = \frac{1}{\cos^2 \theta} = \frac{1}{\frac{1}{4}} = 4\).

### Step 4: Substituting into the Expression

Now, we substitute \(\sin^2 \theta = \frac{3}{4}\) and \(\sec^2 \theta = 4\) into the expression we need to prove:

\[
\frac{\sin^2 \theta - \sec^2 \theta}{\sin^2 \theta + \sec^2 \theta} = \frac{\frac{3}{4} - 4}{\frac{3}{4} + 4}
\]

Simplifying both the numerator and denominator:

\[
\frac{\frac{3}{4} - 4}{\frac{3}{4} + 4} = \frac{\frac{3 - 16}{4}}{\frac{3 + 16}{4}} = \frac{\frac{-13}{4}}{\frac{19}{4}} = \frac{-13}{19}
\]

Thus, the final result is:

\[
\boxed{-\frac{13}{19}}
\]

Let me know if you would like more details or have questions!

---

Here are 5 related questions for practice:

1. What is the value of \(\tan \theta\) given that \(\cot \theta = \frac{1}{\sqrt{3}}\)?
2. How can we derive the identity \(\sec^2 \theta = 1 + \tan^2 \theta\)?
3. What is the value of \(\cos \theta\) when \(\sin \theta = \frac{\sqrt{3}}{2}\)?
4. How do we solve a trigonometric identity involving both \(\sin\) and \(\sec\)?
5. If \(\cot \theta = \frac{1}{\sqrt{3}}\), what is the value of \(\csc \theta\)?

**Tip:** When simplifying trigonometric expressions, always consider using identities such as \(\sin^2 \theta + \cos^2 \theta = 1\) for easier substitutions.

Solve cotθ = 1/√3 and prove (sin²θ - sec²θ) / (sin²θ + sec²θ).

Learn how to solve the trigonometric equation cotθ = 1/√3 and prove the expression (sin²θ - sec²θ) / (sin²θ + sec²θ). This solution covers key trigonometric identities and is suitable for students in Grades 10-12.

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