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Question

f(x)=cos2tan-1sincot-11-xx then


A

(1-x)2f'(x)2(f(x))2=0

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B

(1-x)2f'(x)+2(f(x))2=0

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C

(1+x)2f'(x)2(f(x))2=0

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D

(1+x)2f'(x)+2(f(x))2=0

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Solution

The correct option is B

(1-x)2f'(x)+2(f(x))2=0


Explanation for the correct option:

Step 1. Find the value of given function:

f(x)=cos2tan-1sincot-11-xx

f(x)=cos2tan-1sinsin-1x cot-11-θθ=sin-1θ

f(x)=cos2tan-1x

f(x)=coscos-11-x1+x 2tan-1θ=cos-1(1-θ21+θ2)

f(x)=1-x1+x

Differentiate it with respect to x

f'(x)=-1×(1+x)-1×(1-x)(1+x)2 u(x)v(x)=u'(x).v(x)-v'(x).u(x)v(x)2

=-1-x-1+x1+x2=-21+x2

Step 2. Multiply both sides by 1-x2

(1-x)2f'(x)=-2(1+x)2(1-x)2

(1-x)2f'(x)=-21-x1+x2

(1-x)2f'(x)=-2f(x)2

(1-x)2f'(x)+2(f(x))2=0

Hence, Option ‘B’ is Correct.


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