If `sec theta=sqrt(2)a n d(3pi)/2

Explanation

`sec theta = sqrt2`

As, `(3pi)/2 lt theta le (2pi)`

`:. theta = (7pi)/4`

Now, `(1+tantheta+cosectheta)/(1+cot theta- cosectheta)`

`=(1+tan((7pi)/4)+cosec((7pi)/4))/(1+cot ((7pi)/4)- cosec((7pi)/4))`

`=(1-1-sqrt2)/(1-1+sqrt2)`

`=-sqrt2/sqrt2 = -1`

`:. (1+tantheta+cosectheta)/(1+cot theta- cosectheta) =-1`