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Question

Show that the function f(x)=cot1(sinx+cosx) is decreasing on (0,π4) and increasing on (π4,π2).

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Solution

f(x)=cot1(sinx+cosx)

f(x)=(cosxsinx)1+(sinx+cosx)2=sinxcosx2+sin2x

when
0<x<π/4 0<x<π/4
sinx<cosx
sinxcosx<0

0<2x<π/2
0<sin2x<1
so, 2+sin2x>0 so, f(x)<0 for x(0,π/4)

so f(x) is a decreasing function on interval (0,π/4)
π/4<x<π/2 π/4<x<π/2
π/2<2x<π sinxcosx>0
1<sin2x<1 so, sinxcosx2+sin2x>0

so, 2+sin2x>1

so for x(π/4,π/2) the f(x) is increasing function.

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