Question

Approach

Step-by-step working

1. 1

   LHS $=\frac{\frac{\sin \theta }{\cos \theta }}{1-\frac{\cos \theta }{\sin \theta }}+\frac{\frac{\cos \theta }{\sin \theta }}{1-\frac{\sin \theta }{\cos \theta }}$

   0.5 marks
2. 2

   $=\frac{{\sin }^2\theta }{\cos \theta (\sin \theta -\cos \theta )}-\frac{{\cos }^2\theta }{\sin \theta (\sin \theta -\cos \theta )}$

   1 mark
3. 3

   $=\frac{1}{\sin \theta -\cos \theta }\left[\frac{{\sin }^3\theta -{\cos }^3\theta }{\sin \theta \cos \theta }\right]$

   0.5 marks
4. 4

   $=\frac{(\sin \theta -\cos \theta )({\sin }^2\theta +\sin \theta \cos \theta +{\cos }^2\theta )}{(\sin \theta -\cos \theta )\sin \theta \cos \theta }$

   0.5 marks
5. 5

   $=\frac{1+\sin \theta \cos \theta }{\sin \theta \cos \theta }=1+\sec \theta \csc \theta$ = RHS

   0.5 marks

Concepts used