prove (1+cosx)/(sinx) + (sinx)/(1+cosx) = 2cscx

Question

(1+cosx)/(sinx) + (sinx)/(1+cosx) = 2cscx

use left side as given

Answer 1

Hi Morgan,

 

(1+cosx)/(sinx) + (sinx)/(1+cosx)

make the denominators common by multiplying the first fraction by (1+cosx) and the second fraction by sinx. We get

(1+cosx)(1+cosx)        sinxsinx             1+2cosx+cos^2x + sin^2x       2 + 2cosx           2(1+cosx)            2

---------------------- + -----------------  = -------------------------------- = ------------------ = ----------------- = ------- 

sinx(1+cosx)               (1+cosx)sinx          sinx(1+cosx)                       sinx(1+cosx)        sinx(1+cosx)       sinx

 

2/sinx = 2cscx

 

1+2cosx+cos^2x + sinx^2 = 2 + 2cosx (remember cos^2x+sin^2x = 1)