By writing cotx and tanx in terms of cosx and sinx, show that sinx/(1-cotx) + cosx/(1-tanx) = sinx + cosx.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [M1] OR sinx(1-sinx/cosx) + cosx(1-cosx/sinx) / sinx(1-sinx/cosx) + cosx(1-cosx/sinx) for Writes cotx and tanx in terms of sinx and cosx: sinx/(1-cosx/sinx) + cosx/(1-sinx/cosx)
2. [A1] OR sinx/(sinx-cosx)/sinx + cosx/(cosx-sinx)/cosx for Simplifies denominator: sinx/((sinx-cosx)/sinx) + cosx/((cosx-sinx)/cosx)
3. [A1] OR writes as a single simple algebraic fraction: sin²x(cosx-sinx) + cos²x(sinx-cosx) / (sinx-cosx)(cosx-sinx) for Writes as two simple algebraic fractions: sin²x/(sinx-cosx) + cos²x/(cosx-sinx)
4. [A1] OR sin²x(cosx-sinx) - cos²x(cosx-sinx) / (sinx-cosx)(cosx-sinx) for Writes as a difference with a common denominator: sin²x/(sinx-cosx) - cos²x/(sinx-cosx)
5. [A1] All steps correct and final step fully justified by factorising for Correct simplification to given answer, e.g., (sinx-cosx)(sinx+cosx)/(sinx-cosx) = sinx + cosx
6. [B2] B1 for 9cosx/sinx + 3/sinx = sinx/cosx or better for 10cos²x + 3cosx - 1 [= 0] or sec²x - 3secx - 10 [= 0]
7. [M1] FT their 3-term quadratic in cosx or secx for (5cosx-1)(2cosx+1) [= 0] or (secx-5)(secx+2) [= 0]
8. [A2] A1 for any two correct angles for cosx = 1/5 and cosx = -1/2
9. [A1] 78.5 or 78.46[30...] rot to 2 or more dp, 281.5 or 281.53[69...] rot to 2 or more dp, 120, 240 and no extras in range 0 < x < 360