Question : If $\sin \theta+\cos \theta=\frac{1}{29}$, then find the value of $\frac{\operatorname{sin} \theta+\operatorname{cos} \theta}{\operatorname{sin} \theta-\operatorname{cos} \theta}$.
Option 1: $\frac{1}{41}$
Option 2: $\frac{43}{29}$
Option 3: $\frac{41}{29}$
Option 4: $\frac{1}{43}$
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Correct Answer: $\frac{1}{41}$
Solution : Given: $\sin \theta+\cos \theta=\frac{1}{29}$ We know that $\sin \theta+\cos \theta=\frac{1}{29}$ Squaring both sides, $(\sin \theta+\cos \theta)^2=\frac{1}{29^2}$ ⇒ $1+2\sin \theta\cos \theta=\frac{1}{841}$ ⇒ $2\sin \theta\cos \theta=\frac{1}{841}-1=-\frac{840}{841}$...(i) Now, $(\sin \theta-\cos \theta)^2=1-2\sin \theta\cos \theta$ From equation (i) $(\sin \theta-\cos \theta)^2=1-(-\frac{840}{841})=\frac{1681}{841}$ ⇒ $(\sin \theta-\cos \theta)=\sqrt{(\frac{1681}{841})}=\frac{41}{29}$ ⇒ $(\sin \theta-\cos \theta) =\frac{41}{29}$ ⇒ $\frac{\operatorname{sin} \theta+\operatorname{cos} \theta}{\operatorname{sin} \theta-\operatorname{cos} \theta}=\frac{1}{29}×\frac{29}{41}=\frac{1}{41}$ Hence, the correct answer is $\frac{1}{41}$.
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