Question

Approach

Step-by-step working

1. 1

   LHS = $\frac{1+{\tan }^2\theta }{{\tan }^2\theta }+\frac{1+{\cot }^2\theta }{{\cot }^2\theta }=\frac{{\sec }^2\theta }{{\tan }^2\theta }+\frac{{\csc }^2\theta }{{\cot }^2\theta }$.

   1 mark
2. 2

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

   1 mark
3. 3

   $=\frac{{\cos }^2\theta +{\sin }^2\theta }{{\sin }^2\theta {\cos }^2\theta }=\frac{1}{{\sin }^2\theta (1-{\sin }^2\theta )}=\frac{1}{{\sin }^2\theta -{\sin }^4\theta }$ = RHS.

   1 mark

Concepts used