Question

Approach

Step-by-step working

1. 1

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

   0.5 marks
2. 2

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

   0.5 marks
3. 3

   $=\frac{{\sin }^3\theta -{\cos }^3\theta }{\sin \theta \cos \theta (\sin \theta -\cos \theta )}$.

   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

   $=\frac{{\sin }^2\theta +{\cos }^2\theta +\sin \theta \cos \theta }{\sin \theta \cos \theta }=\frac{1+\sin \theta \cos \theta }{\sin \theta \cos \theta }$.

   0.5 marks
6. 6

   $=\frac{{\sin }^2\theta }{\sin \theta \cos \theta }+\frac{{\cos }^2\theta }{\sin \theta \cos \theta }+\frac{\sin \theta \cos \theta }{\sin \theta \cos \theta }=\tan \theta +\cot \theta +1=$ RHS.

   0.5 marks

Concepts used