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a
(tanx)ln(sinx)>(cotx)ln(sinx),∀x∈(0,π/4)
b
4lncosecx<5lncosecx,∀x∈(0,π/2)
c
(1/2)ln(cosx)<(1/3)ln(cosx),∀x∈(0,π/2)
d
2ln(tanx)>2ln(sinx),∀x∈(0,π/2)
answer is A.
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Detailed Solution
1 For x∈0,π4,tanx(cotx)ln(sinx)2 For x∈0,π2,cosecx≥1 ⇒ ln(cosecx)≥0 ⇒ 4ln(cosecx)<5ln(cosecx)3 x∈0,n2 ⇒cosx∈(0,1) ⇒ ln(cosx)<0 also 12>13 ⇒−12ln(cosx)<13ln(cosx)4 For x∈0,π2 Since sinx < tanx, we get ln(sinx) < ln(tanx) ⇒ 2ln(sinx)<2ln(tanx)