**Cos theta- sin theta+1/cos theta+ sin theta -1 = cosec theta +cot theta**

Sol. L.H.S. Cos-Sin+1Cos+Sin-1

Cos-Sin+1Cos+Sin-1Cos+Sin+1Cos+Sin+1

(Cos+1)-Sin(Cos+Sin)-1(Cos+1)+Sin(Cos+Sin)+1 Because (a-b)(a+b)=a2-b2

(Cos+1)^2-Sin^2(Cos+Sin)^2-1

Cos^2+1+2Cos-Sin2Cos2+Sin2+2SinCos-1

Cos2+2Cos+(1-Sin2)(Cos2+Sin2)+2SinCos-1

Cos2+2Cos+Cos21+2SinCos-1

2Cos2+2Cos2SinCos

2Cos(1+Cos)2SinCos

1+CosCos

1Sin+CosSinCosec+Cot=R.H.S.