Question

If \(\sin \theta - \cos \theta = \frac{4}{5}\) then the value of \(\sin \theta + \cos \theta =\)

• A. \(\frac{5}{\sqrt{34}}\)
• B. \(-\frac{5}{\sqrt{34}}\)
• C. \(\frac{\sqrt{34}}{25}\)
• D. \(\frac{\sqrt{34}}{5}\)

Hint:

The identity \((\sin\theta \pm \cos\theta)^2 = 1 \pm 2\sin\theta\cos\theta\) is extremely useful. Memorizing the combined identity \((\sin\theta - \cos\theta)^2 + (\sin\theta + \cos\theta)^2 = 2\) provides a direct shortcut for problems where one expression is given and the other is asked.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the value of sin θ - cos θ and we need to find the value of sin θ + cos θ.

Step 2: Key Formula or Approach (Alternate Method):
Use the identity (sin θ - cos θ)² + (sin θ + cos θ)² = 2 directly, avoiding expansion of squares.

Step 3: Detailed Explanation:
Let x = sin θ + cos θ. Given sin θ - cos θ = 4/5. Using identity: (sin θ - cos θ)² + (sin θ + cos θ)² = 2(sin²θ + cos²θ) = 2(1) = 2. Substitute: (4/5)² + x² = 2 → 16/25 + x² = 2 → x² = 2 - 16/25 = (50-16)/25 = 34/25 → x = √34/5 (taking positive value).

Step 4: Final Answer:
The value of sin θ + cos θ is √34/5.