Question : $\theta$ is a positive acute angle and $\sin\theta-\cos\theta=0$, then the value of $\sec\theta+\operatorname{cosec}\theta$ is:
Option 1: $2$
Option 2: $\sqrt{3}$
Option 3: $2\sqrt{2}$
Option 4: $3\sqrt{2}$
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Correct Answer: $2\sqrt{2}$
Solution :
Given:
$\sin\theta-\cos\theta=0$
⇒ $\sin\theta=\cos\theta$
⇒ $\sin\theta=\sin\;(90°-\theta)$
⇒ $2\theta=90°$
⇒ $\theta=45°$
So, $\sec\theta+\operatorname{cosec}\theta$
$= \sec 45°+ \operatorname{cosec} 45°$
$= \sqrt{2}+\sqrt{2}$
$= 2\sqrt{2}$
Hence, the correct answer is $2\sqrt{2}$.
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