Question : Which of the following is equal to $[\frac{\tan \theta+\sec \theta–1}{\tan \theta–\sec \theta+1}]$?
Option 1: $\frac{1+\sin \theta}{\cos \theta}$
Option 2: $\frac{1+\tan \theta}{\cot \theta}$
Option 3: $\frac{1+\cot \theta}{\tan \theta}$
Option 4: $\frac{1+\cos \theta}{\sin \theta}$
Correct Answer: $\frac{1+\sin \theta}{\cos \theta}$
Solution :
Given: The given trigonometric expression is $[\frac{\tan \theta+\sec \theta–1}{\tan \theta–\sec \theta+1}]$.
Use the trigonometric identity, $\sec^2\theta–\tan^2\theta=1$
⇒ $[\frac{\tan \theta+\sec \theta–(\sec^2\theta–\tan^2\theta)}{\tan \theta–\sec \theta+1}]=[\frac{(\tan \theta+\sec \theta)(1–\sec\theta+\tan \theta)}{\tan \theta–\sec \theta+1}$
$=\tan \theta+\sec \theta=\frac{1+\sin \theta}{\cos \theta}$
Hence, the correct answer is $\frac{1+\sin \theta}{\cos \theta}$.
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