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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
a. For, x∈0,π4,tanx(cotx)ln(sinx)b. For x∈0,π2,cosecx≥1 ⇒ ln(cosecx)≥0⇒ 4ln(cosecx)<5ln(cosecx)c. x∈0,π2⇒cosx∈(0,1)⇒ ln(cosx)<0 Also, 12>13⇒12ln(cosx)<13ln(cosx)d. For x∈0,π2 Since sin x < tan x, we get ln(sin x) < ln(tan x) ⇒ 2ln(sinx)<2ln(tanx)