f(x) = \(cos((2 tan^{-1}((sin((cot^{-1}(\sqrt{((1-x)/x) )} ) )\) then

a. (1-x)2 f'(x) – 2(f(x)) 2 = 0

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

c. (1+x) 2 f'(x) – 2(f(x)) 2= 0

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

Solution:

Answer: (b)

f(x) = [latex]cos((2 tan^{-1}((sin((cot^{-1}(\sqrt{((1-x)/x) )} ) )[/latex] then

f(x) = [latex]cos((2 tan^{-1}((sin((sin^{-1}(\sqrt{x} ) ) ) [/latex]

= cos((2 tan-1(√x)

= cos((cos-1 (((1 – x)/(1 + x)) ) = (1 – x)/(1 + x)

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

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

(1 – x) 2 f'(x) + 2(f(x)) 2= 0

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