Question : If $\sin \theta-\cos \theta=\frac{4}{5}$, then find the value of $\sin \theta+\cos \theta$.
Option 1: $ \frac{5}{\sqrt{34}} $
Option 2: $ \frac{5}{\sqrt{24}} $
Option 3: $ \frac{\sqrt{34}}{5} $
Option 4: $ \frac{\sqrt{24}}{5}$
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Correct Answer: $ \frac{\sqrt{34}}{5} $
Solution :
$\sin \theta-\cos \theta=\frac{4}{5}$
⇒ $(\sin \theta-\cos \theta)^2=(\frac{4}{5})^2$
⇒ $\sin^2 \theta + \cos^2 \theta -2\sin \theta \cos \theta = \frac{16}{25}$
⇒ $1-2\sin \theta \cos \theta = \frac{16}{25}$
⇒ $2\sin \theta \cos \theta = \frac{25-16}{25}$
⇒ $2\sin \theta \cos \theta = \frac{9}{25}$ -------------(i)
Adding $\sin^2 \theta + \cos^2 \theta$ on both sides of equation (i),
$\sin^2 \theta + \cos^2 \theta +2\sin \theta \cos \theta = \sin^2 \theta + \cos^2 \theta+ \frac{9}{25}$
⇒ $(\sin \theta+\cos \theta)^2= 1+\frac{9}{25}$
⇒ $\sin \theta+\cos \theta = \sqrt{\frac{25+9}{25}} = \frac{\sqrt{34}}{5}$
Hence, the correct answer is $\frac{\sqrt{34}}{5}$.
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