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Question

If -π2<x<π2, then the value of log(secx) is


A

2coth-1cosec2x2-1

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B

2coth-1cosec2x2-1

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C

2cosech-1cot2x2-1

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D

2cosech-1cot2x2+1

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Solution

The correct option is A

2coth-1cosec2x2-1


Explantion for the correct option:

Step 1: Apply the given condition -π2<x<π2

Let us consider, log(secx)=p

secx=ep

1cosx=ep

Write ep=ep2+p2 in the above equation

1cosx=ep2+p2

1cosx=ep2e-p2

Step 2: Apply component - dividend rule,

1+cosx1-cosx=ep2+e-p2ep2-e-p2 1+cosx=2cos2x2and1-cosx=2sin2x2

2cos2x22sin2x2=ep2+e-p2ep2-e-p2

cot2x2=cothp2 cothp2=ep2+e-p2ep2-e-p2

From the above equation, we can write

cothp2=cot2x2

p2=coth-1cot2x2

p=2coth-1cosec2x2-1 ; cot2x2=cosec2x2-1

log(secx)=2coth-1cosec2x2-1 ; log(secx)=p

Therefore, the value of log(secx)=2coth-1cosec2x2-1.

Hence, the correct answer is an option (A).


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