Question

If sin \(\theta\) + cos \(\theta\) = \(\sqrt{3}\), then prove that tan \(\theta\) + cot \(\theta\) = 1.

Hint:

A useful identity to remember:
\[ \tan \theta + \cot \theta = \frac{1}{\sin \theta \cos \theta} \]
Remembering this standard conversion helps you link the two parts of the question directly and reduces intermediate algebraic steps.

Answer 1

Step 1: Understanding the Question:
The topic of this question is Trigonometric Identities.
We are given an equation, \( \sin \theta + \cos \theta = \sqrt{3} \), and we need to use algebraic and trigonometric identities to prove that \( \tan \theta + \cot \theta = 1 \).

Step 2: Key Formula or Approach:
- Squaring both sides of the given equation to find the value of the product \( \sin \theta \cos \theta \).
- Use the fundamental trigonometric identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
- Express \( \tan \theta \) and \( \cot \theta \) in terms of sine and cosine:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]

Step 3: Detailed Explanation:
1. Start with the given equation:
\[ \sin \theta + \cos \theta = \sqrt{3} \]
2. Square both sides of the equation:
\[ (\sin \theta + \cos \theta)^2 = (\sqrt{3})^2 \]
3. Expand the left-hand side using the algebraic identity \( (a + b)^2 = a^2 + b^2 + 2ab \):
\[ \sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta = 3 \]
4. Substitute the standard identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ 1 + 2 \sin \theta \cos \theta = 3 \]
5. Solve for the product \( \sin \theta \cos \theta \):
\[ 2 \sin \theta \cos \theta = 3 - 1 \]
\[ 2 \sin \theta \cos \theta = 2 \]
\[ \sin \theta \cos \theta = 1 \quad \text{(Equation 1)} \]
6. Now, consider the expression we need to evaluate:
\[ \text{LHS} = \tan \theta + \cot \theta \]
7. Express \( \tan \theta \) and \( \cot \theta \) in terms of sine and cosine:
\[ \text{LHS} = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \]
8. Take the common denominator to combine the terms:
\[ \text{LHS} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} \]
9. Substitute \( \sin^2 \theta + \cos^2 \theta = 1 \) into the numerator:
\[ \text{LHS} = \frac{1}{\sin \theta \cos \theta} \]
10. Substitute the value of \( \sin \theta \cos \theta = 1 \) from Equation 1:
\[ \text{LHS} = \frac{1}{1} = 1 = \text{RHS} \]
This completes the proof.

Step 4: Final Answer:
Hence, it is proved that \(\tan \theta + \cot \theta = 1\) when \(\sin \theta + \cos \theta = \sqrt{3}\).