Question

If \(\sin\theta+\cos\theta=\sqrt{2}\), then \(\tan\theta+\cot\theta\) equals:

• A. 1
• B. 2
• C. \(\sqrt{2}\)
• D. 0

Hint:

Maximum value of \(\sin\theta+\cos\theta\) is \(\sqrt{2}\).

Answer 1

The Correct Option is B

To solve the given problem, we need to find the value of \( \tan\theta + \cot\theta \) given that \( \sin\theta + \cos\theta = \sqrt{2} \).

1. We start by squaring the given equation: \[ (\sin\theta + \cos\theta)^2 = (\sqrt{2})^2 \] This simplifies to: \[ \sin^2\theta + 2\sin\theta\cos\theta + \cos^2\theta = 2 \]
2. Recall the Pythagorean identity: \[ \sin^2\theta + \cos^2\theta = 1 \]
3. Substitute this identity into the equation obtained in step 1: \[ 1 + 2\sin\theta\cos\theta = 2 \]
4. Simplify the equation to find: \[ 2\sin\theta\cos\theta = 1 \] which implies \[ \sin\theta\cos\theta = \frac{1}{2} \]
5. Now, express \( \tan\theta + \cot\theta \) in terms of \( \sin\theta \) and \( \cos\theta \): \[ \tan\theta = \frac{\sin\theta}{\cos\theta} \quad \text{and} \quad \cot\theta = \frac{\cos\theta}{\sin\theta} \]
6. Therefore, \[ \tan\theta + \cot\theta = \frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} \]
7. Combine the fractions: \[ \tan\theta + \cot\theta = \frac{\sin^2\theta + \cos^2\theta}{\sin\theta\cos\theta} \]
8. Substitute the known values:
   • \( \sin^2\theta + \cos^2\theta = 1 \) (Pythagorean identity)
   • \( \sin\theta\cos\theta = \frac{1}{2} \) (from step 4)
9. This results in: \[ \tan\theta + \cot\theta = \frac{1}{\frac{1}{2}} = 2 \]

Therefore, the value of \( \tan\theta + \cot\theta \) is 2, which matches the given correct answer.