Question : $(\sin \theta+\operatorname{cosec} \theta)^2+(\cos \theta+\sec \theta)^2=$?
Option 1: $5+\tan ^2 \theta+\cot ^2 \theta$
Option 2: $7+\tan ^2 \theta-\cot ^2 \theta$
Option 3: $7+\tan ^2 \theta+\cot ^2 \theta$
Option 4: $5+\tan ^2 \theta-\cot ^2 \theta$
Correct Answer: $7+\tan ^2 \theta+\cot ^2 \theta$
Solution :
$(\sin \theta+\operatorname{cosec} \theta)^2+(\cos \theta+\sec \theta)^2$
= $(\sin^2 \theta+\operatorname{cosec}^2 \theta+2\sin \theta\operatorname{cosec} \theta)+(\cos^2 \theta+\sec^2 \theta+2\cos \theta\sec \theta)$
= $\sin^2 \theta+\cos^2 \theta+\operatorname{cosec}^2 \theta+2\sin \theta\operatorname{cosec} \theta+\sec^2 \theta+2\cos \theta\sec \theta$
= $1+1+\cot^2 \theta+2+1+\tan^2 \theta+2$
= $7+\cot^2 \theta+\tan^2 \theta$
Hence, the correct answer is $7+\tan^2 \theta+\cot^2 \theta$.
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