Question : $\frac{\sin ^2 \theta}{\cos \theta(1+\cos \theta)}+\frac{1+\cos \theta}{\cos \theta}=$?
Option 1: $\operatorname{cosec}\theta$
Option 2: $\sec\theta$
Option 3: $2\cos\theta$
Option 4: $2\sec\theta$
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Correct Answer: $2\sec\theta$
Solution :
$\frac{\sin ^2 \theta}{\cos \theta(1+\cos \theta)}+\frac{1+\cos \theta}{\cos \theta}$
$=\frac{1-\cos ^2 \theta}{\cos \theta(1+\cos \theta)}+\frac{1+\cos \theta}{\cos \theta}$
$=\frac{(1-\cos \theta)(1+\cos \theta)}{\cos \theta(1+\cos \theta)}+\frac{1+\cos \theta}{\cos \theta}$
$=\frac{1-\cos \theta}{\cos \theta}+\frac{1+\cos \theta}{\cos \theta}$
$=\frac{1-\cos \theta+1+\cos \theta}{\cos \theta}$
$=\frac{2}{\cos \theta}$
$=2\sec\theta$
Hence, the correct answer is $2\sec\theta$.
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