Show that sinx / (1-cosx) + (1-cosx) / sinx = 2 cosecx.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

1. [M1] (sin²x + (1-cosx)²) / ((1-cosx)sinx)
2. [A1] (sin²x + 1 - 2cosx + cos²x) / ((1-cosx)sinx)
3. [A1] (1 + 1 - 2cosx) / ((1-cosx)sinx)
4. [A1] (2 - 2cosx) / ((1-cosx)sinx) = 2(1-cosx) / ((1-cosx)sinx) = 2/sinx = 2 cosec x (All steps correct and final step justified)
5. [M1] (sin x(1+cos x))/((1-cosx)(1+cosx)) + ((1-cosx)sin x)/(sin x sin x) (Alternative 1)
6. [A1] (sin x(1+cos x)) / (1-cos²x) + ((1-cosx)sin x) / sin²x (Alternative 1)
7. [A1] (sin x + sin x cos x) / sin²x + (sin x - cos x sin x) / sin²x (Alternative 1)
8. [A1] (2sin x) / sin²x = 2/sinx = 2 cosec x (All steps correct and final step justified, Alternative 1)
9. [B1] 3sin²x - sinx - 2 [= 0] soi
10. [M1] (3sinx + 2)(sinx - 1) [= 0] oe
11. [A1] sinx = -2/3
12. [A1] sinx = 1
13. [A1] 90
14. [A1] 221.8 or 221.81[03...] rot to 2 or more dp
15. [A1] 318.2 or 318.18[96...] rot to 2 or more dp
16. [A1] and no extras in range