Question

Please some one prove : cot A-cos A / cot A+cosA = cosec A-1/ cosec A+1

Answer 1

L.H.S=(cot A-cos A)/(cot A+cos A)
         =[(cos A/sin A)-cos A]/[(cos A/sin A)+cos A]
         =[(cos A-cos A×sin A)/sin A]/[(cos A+cos A×sin A)/sin A]
         =[cos A(1-sin A)]/[cos A(1+sin A)]
         =(1-sin A)/(1+sin A)

R.H.S=(cosec A-1)/(cosec A+1)
         =[(1/sin A)-1]/[(1/sin A)+1]
         =[(1-sin A)/sin A]/[(1+sin A)/sin A]
         =(1-sin A)/(1+sin A)

L.H.S=R.H.S