Question

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

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

Hint:

Use \((a+b)^2+(a-b)^2=2(a^2+b^2)\) for quick calculation.

Answer 1

The Correct Option is D

Step 1: Given, \[ \sin\theta-\cos\theta=\frac{4}{5} \]

Step 2: Square both sides: \[ (\sin\theta-\cos\theta)^2=\frac{16}{25} \] \[ \sin^2\theta+\cos^2\theta-2\sin\theta\cos\theta=\frac{16}{25} \] \[ 1-2\sin\theta\cos\theta=\frac{16}{25} \]

Step 3: \[ 2\sin\theta\cos\theta=1-\frac{16}{25}=\frac{9}{25} \]

Step 4: Now, \[ (\sin\theta+\cos\theta)^2 = \sin^2\theta+\cos^2\theta+2\sin\theta\cos\theta \] \[ =1+\frac{9}{25} \] \[ =\frac{34}{25} \]

Step 5: Therefore, \[ \sin\theta+\cos\theta=\frac{\sqrt{34}}{5} \] \[ \boxed{\frac{\sqrt{34}}{5}} \]