3.81: Cos/sin identity to cosec

0606/22/O/N/22 — Question 7 · 8 marks

(a)
Show that cosx1sinx+sinx1cosx=cosecx\dfrac{\mathrm{cos}\,x}{1-\mathrm{sin}\,x}+\dfrac{\mathrm{sin}\,x}{1-\mathrm{cos}\,x}=\mathrm{cosec}\,x.
[4]
(b)
Hence solve the equation cosx1sinx+sinx1cosx=2\dfrac{\mathrm{cos}\,x}{1-\mathrm{sin}\,x}+\dfrac{\mathrm{sin}\,x}{1-\mathrm{cos}\,x}=2 in the interval given on the paper.
[4]