# prove that 1/(cosecA-cotA) - 1/sinA = 1/sinA - 1/(cosecA+cotA)

Rinkoo Gupta
9 years ago
LHS=1/(cosecA-cotA) -1/sinA
=(cosecA+cotA)/cosec^2A-cot^2A) -cosecA
=cosecA+cotA-cosecA
=cotA
RHS=1/sinA-1/(cosecA+cotA)
=cosecA-(cosecA-cotA)/(cosec^2A-cot^2A)
=cosecA-(cosecA-cotA)
=cotA
LHS=RHS
Thanks & Regards
Rinkoo Gupta
Purendar
19 Points
6 years ago
1/cosecA-cotA - 1/sinA= 1/sinA - 1/cosecA+cotA Taking LHS 1/(cosecA-cotA )- 1/sinA 1/(cosecA-cotA ) *(cosecA+cotA )/(cosecA +cotA )- 1/sinA (rationalizing the denominator) (cosecA +cotA )/(cosec^2A +cot^2A )-1/sinA As (cosec^2A +cot^2A )=1 (cosecA +cotA ) - 1/sinA 1/sinA +(cotA-cosecA) 1/sinA +(cotA-cosecA)*(cosecA+cotA )/(cosecA+cotA ) (ratioalizing the numerator) 1/sinA +(cot^2A-cosec^2A)/(cosecA+cotA ) 1/sinA +1/(cosecA+cotA ) As (cosec^2A +cot^2A )=1 =RHS
Sipra Nayak
13 Points
4 years ago
1/(cosecA-cotA) -1/sinA
=(Cosec^2A-cot^2A)/(cosecA-cotA)-cosecA .  (As cosec^2A-cot^2A=1)
=(CosecA+cotA)-cosecA
=CosecA-(CosecA-cotA)
=1/sinA-(cosecA-cotA)(cosecA+cotA)/(cosecA+cotA)
=1/sinA-(cosec^2A-cot^2A)/(cosecA+cotA)
=1/sinA-1/(cosecA+cotA)=RHS. Proved