Question

Prove that (sin a/1+cos a + 1+cosa/sin a) (sin a /1-cos a - 1-cos a /sin a) =4cosec a/cot a

Answer 1

(sin A/1+cos A + 1+cos A/ sin A) ( sin A/1- cos A - 1-cos A/ sin A)
= {sin2 A + (1+ cos A)2/(1+cos A) (sin A)} { sin2A - (1- cos A)2/(1-cos A) (sin A)} [ I took LCM here...seperately okay?]
={sin2 A+1+2cos A+cos2A/(1+cos A)(sin A)} {sin2 A- (1- 2cos A +cos2 A)/(1-cos A) (sin A)} [Now, I expanded using (a+b)2 and (a-b)2]
={1+1+2cos A/(1+cos A) (sin A)} {sin2 A- 1+2cos A -cos2 A/(1-cos A) (sin A)} [ Substitute sin2 A+cos2 A]
={2+2cos A/(1 + cos A)(sin A)} {1-cos2 A- cos2 A-1+2cos A/(1-cos A) (sin A)} [substituted sin2 A]
={2 (1+cosA)/ (1+cos A) (sin A)} {2cos A (-cos A+1)/ (1-cos A)(sin A)}
[ Simplify, Simplify and Simplify..Cut off those possible..]
= {2/sin A}{2cos A/sin A}
=2cosec A* 2 cot A
=4cosec A*cot A 

Answer 2

i solved in this pic. please take a look. hope it helps