Question : If $\cos \theta+\sin \theta=\sqrt{2}$, then what is the value of $\sec \theta \operatorname{cosec} \theta$ ?
Option 1: $\frac{1}{2}$
Option 2: $1$
Option 3: $2$
Option 4: $0$
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Correct Answer: $2$
Solution :
Given: $\cos \theta+\sin \theta=\sqrt{2}$
Squaring both sides, we get:
⇒ $(\cos \theta+\sin \theta)^2=(\sqrt{2})^2$
⇒ $\cos^2 \theta+\sin^2 \theta+2\sin\theta \cos\theta=2$
We know that $\cos^2 \theta+\sin^2 \theta = 1$
Thus, $1+2\sin\theta \cos\theta=2$
⇒ $2\sin\theta \cos\theta=2-1$
⇒ $\sin\theta \cos\theta=\frac{1}{2}$
⇒ $\frac{1}{\operatorname{cosec}\theta}\frac{1}{\sec\theta}=\frac{1}{2}$
⇒ $\operatorname{cosec}\theta \sec\theta = 2$
Hence, the correct answer is $2$.
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