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Which of the following inequalities are correct? (A) (tan x)^(ln sin x) > (cot x)^(ln sin x) for all x in (0, pi/4) (B) 4^(ln cosec x) < 5^(ln cosec x) for all x in (0, pi/2) (C) (1/e)^(ln cos x) < (1/3)^(ln cos x) for all x in (0, pi/2) (D) 2^(log_(1/2)(tan x)) < 2^(log_(1/2)(sin x)) for all x in (0, pi/2)

1. A and B only
2. A, B and C only
3. A, C and D only
4. A, B, C and D

Correct answer: A, B, C and D

Solution

(A) In (0,pi/4): tan x < 1 < cot x, and ln(sin x) < 0. With a negative exponent, the larger base gives the smaller value, so (tan x)^p > (cot x)^p. TRUE. (B) ln(cosec x) > 0 for x in (0,pi/2) and 4 < 5, so 4^p < 5^p. TRUE. (C) ln(cos x) < 0 for x in (0,pi/2); 1/e > 1/3, with negative exponent (1/e)^p < (1/3)^p. TRUE. (D) log_(1/2)(tan x) < log_(1/2)(sin x) since tan x > sin x and log base 1/2 is decreasing; so 2^(smaller) < 2^(larger). TRUE.

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