Question : If $\sin \theta-\cos \theta=\frac{1}{5}$, then find the value of $\sin \theta+\cos \theta$.
Option 1: $\frac{5}{7}$
Option 2: $\frac{7}{5}$
Option 3: $\frac{5}{3}$
Option 4: $\frac{3}{5}$
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Correct Answer: $\frac{7}{5}$
Solution :
Given, $\sin \theta-\cos \theta=\frac{1}{5}$
Squaring both sides, we get,
$\sin^2 \theta+\cos^2 \theta -2\sin \theta \cos \theta=\frac{1}{25}$
$⇒1-2\sin \theta \cos \theta=\frac{1}{25}$
$⇒2\sin \theta \cos \theta=\frac{24}{25}$
Adding 1 with both sides, we get,
$⇒\sin^2 \theta+\cos^2 \theta +2\sin \theta \cos \theta=1+\frac{24}{25}$ [$\because \sin^2 \theta+\cos^2 \theta=1$]
$⇒(\sin \theta+\cos \theta)^2=\frac{49}{25}$
$⇒\sin \theta+\cos \theta = \frac{7}{5}$
Hence, the correct answer is $\frac{7}{5}$.
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