Question

Prove that : cos theta -sin theta +1/ cos theta +sin theta -1 = cosec theta +cot a

Answer 1

 cos A - sin A+1/cos A + sin A - 1= cosec A+cot A
Solution:

LHS:
cos A - sin A+1/cos A + sin A - 1
=cos A - sin A+1/cos A + sin A - 1* cos A + sin A +1/cos A + sin A +1
                                          [taking conjugate]
=cos^2 A+cos A sin A + cos A - sin A cos A - sin^2 A - sin A + cos A +sin A+1/cos^2 A + cos A sin A +cos A  +sin A cos A + sin A +sin A - cos A - sin A - 1
=cos^2 A - sin^2 A + 2 cos A + 1/2 sin A cos A 
=2 cos^2 A +2 cos A /2 sin A cos A
=2 cos A (cos A + 1) / 2 sin A cos A
=1 + cos A / sin A

RHS:
cosec A+cot A
cosec A+cot A= 1/ sin A + cos A / sin A
=1 + cos A / sin A
                          
                        Therefore, LHS = RHS
                                Hence proved.