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Question

Which of the following option is always correct for x(0,π2)?

A
sin(110)(110)<sin(19)(19)
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B
sin(110)(19)>sin(19)(110)
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C
sin(110)(110)>sin(19)(19)
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D
sin(110)(19)=sin(19)(110)
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Solution

The correct option is C sin(110)(110)>sin(19)(19)
Let f(x)=sinxx
f(x)=xcosxsinxx2
f(x)=(cosxx2)(xtanx)
f(x)<0 x (0,π2)
(x<tanx in(0,π2)).
Hence f(x) is strictly decreasing
We know, (110)<(19)
f(110)>f(19)
sin(110)(110)>sin(19)(19)

Let f(x)=xsinx
f(x)=xcosx+sinx=cosx(x+tanx)
cosx,tanx>0 in x(0,π2)
f(x)>0,x(0,π2)
f(x) strictly increases
for π2>x>0,f(x)>f(0)
19sin(19)>110sin(110)
sin(110)(19)<sin(19)(110)

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