Sove `[cot^(-1) x] + [cos^(-1) x] =0`, where `[.]` denotes the greatest integer function

Sove `[cot^(-1) x] + [cos^(-1) x] =0`, where `[.]` denotes the greates

1.	Sove `[cot^(-1) x] + [cos^(-1) x] =0`, where `[.]` denotes the greatest integer function
Answer» We have `[cot^(-1) x] + [cos^(-1) x] =0` Since `cos^(-1)x, cot^(-1) x ge 0`, we must have `[cot^(-1) x] = [cos^(-1) x] =0` Now, `[cot^(-1) x] = 0` `rArr 0 lt cot^(-1) x lt 1` `rArr x in (cot 1, oo)`...(1) Also, `[cos^(-1) x] = 0` ltrbgt `rArr 0 le cos^(-1) x lt1` `rArr x in (cos 1, 1]`...(2) From (1) and (2) `x in (cot 1,10`

`tan^(-1)[(cosx)/(1+sinx)]`is equal to`pi/4-x/2,forx in (-pi/2,(3pi)/2)``pi/4-x/2,forx in (-pi/2,pi/2)``pi/4-x/2,forx in (-pi/2,pi/2)``pi/4-x/2,forx in (-(3pi)/2,pi/2)`
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