Question : Which of the following is equal to $\frac{1}{\tan \theta}+\tan \theta$?
Option 1: $\sec \theta \operatorname{cosec} \theta$
Option 2: $1$
Option 3: $\frac{\operatorname{cosec} \theta}{\sec \theta}$
Option 4: $\tan ^2 \theta$
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Correct Answer: $\sec \theta \operatorname{cosec} \theta$
Solution :
Given: The trigonometric expression is $\frac{1}{\tan \theta}+\tan \theta$.
We know the trigonometric identity $\sec^2\theta=1+\tan^2\theta$
⇒ $\frac{1}{\tan \theta}+\tan \theta=\frac{1+\tan^2 \theta}{\tan \theta}$
$=\frac{\sec^2 \theta}{\tan \theta}=\frac{\sec^2 \theta}{\sin\theta}\times\cos\theta=\sec \theta \operatorname{cosec} \theta$
Hence, the correct answer is $\sec \theta \operatorname{cosec} \theta$.
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