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prove that tan^-1(cosx/(1-sinx))-cot^-1(...

prove that `tan^-1(cosx/(1-sinx))-cot^-1(sqrt(1+cosx)/sqrt(1-cosx))=pi/4 , xepsilon(0,pi/2)`

Text Solution

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`tan^(-1)A -tan^(-1)B=tan^(-1)((A-B)/(1+AB))`.......(1)
Also `cot^(-1)x=tan^(-1)(1/x)`
`tan^(-1)(cosx/(1-sinx))-tan^(-1)sqrt(((1-cosx)(1-cosx))/((1+cosx)(1-cosx))`
`tan^(-1)(cosx/(1-sinx))-tan^(-1)((1-sinx)/cosx)`
using (1) we get `tan^(-1)1=pi/4`
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