2\ +\ tan²\ 2A\ +\ cot²\ 2A over sec\ 2A\ .\ cosec\ 2A is equal to ____.

2\ +\ tan²\ 2A\ +\ cot²\ 2A over sec\ 2A\ .\ cosec\ 2A is…

${2\ +\ tan^2\ 2A\ +\ cot^2\ 2A} \over sec\ 2A\ .\ cosec\ 2A$ is equal to ____________.

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Correct answer: D

Explanation

The correct answer is Option 4 i.e. sec 2A.cosec 2A.

1 + tan²A = sec²A

1 + cot²A = cosec²A

sec²A + cosec²A = (1/cos²A) + (1/sin²A) = 1/(sin²A.cos²A) = sec²A.cosec²A

${2\ +\ tan^2\ 2A\ +\ cot^2\ 2A} \over sec\ 2A\ .\ cosec\ 2A$

⇒ ${1\ +\ tan^2\ 2A\ +1+\ cot^2\ 2A} \over sec\ 2A\ .\ cosec\ 2A$

⇒ ${sec^22A +cosec^22A} \over sec\ 2A\ .\ cosec\ 2A$

⇒ ${sec^2 2A\ .\ cosec^22A} \over sec\ 2A\ .\ cosec\ 2A$

⇒ sec 2A.cosec 2A