Prove that  sinA/(1+cosA) + (1+cosA)/sinA = 2/sinA

Expert Answers
lfryerda eNotes educator| Certified Educator

To prove a trigonometric identity, you need to start from one side of the identity and, using standard formulas, change it to the other side of the identity.  Generally, you start from the side that is more complicated.

`LS = sinA/(1+cosA)+(1+cosA)/sinA`   find common denominator

`={sin^2A+(1+cosA)^2}/{sinA(1+cosA)}`   simplify numerator

`={sin^2A+1+2cosA+cos^2A}/{sinA(1+cosA)}`

`={2+2cosA}/{sinA(1+cosA)}`  factor numerator

`={2(1+cosA)}/{sinA(1+cosA)}`   cancel common factor

`=2/sinA`

`=RS`

The identity has been proven.

karnishekhawat | Student

Sin A/(1+cosA) + (1+cosA)/sinA

= sin^2 A +(1+cosA)^2 /sinA(1+cosA)

=sin^2 A + 1+cos^2 A+2 cos A / sin A (1+cos A)

=2+2 Cos A/sinA(1+cos A)

=2(1+cos A)/sin A (1+cos A)

= 2/sin A

Hence proved