(a) Prove that: (1+1/(tan^2 θ))(1+1/(cot^2 θ))=1/(sin^2 θ- sin^4 θ)

\[\begin{aligned} \mathrm{LHS} & =\left(1+\cot ^{2} \theta\right)\left(1+\tan ^{2} \theta\right) \\ & =\operatorname{cosec}^{2} \theta \cdot \sec ^{2} \theta \\ & =\frac{1}{\sin ^{2} \theta} \cdot \frac{1}{\cos ^{2} \theta} \\ & =\frac{1}{\sin ^{2} \theta\left(1-\sin ^{2} \theta\right)} \\ & =\frac{1}{\sin ^{2} \theta-\sin ^{4} \theta}=\mathrm{RHS} \end{aligned} \]

\[\begin{aligned} \mathrm{LHS} & =\frac{\operatorname{cosec} \theta-1+\operatorname{cosec} \theta+1}{\sqrt{(\operatorname{cosec} \theta+1)(\operatorname{cosec} \theta-1)}} \\ & =\frac{2 \operatorname{cosec} \theta}{\sqrt{\left(\operatorname{cosec}^{2} \theta-1\right)}} \\ & =\frac{2 \operatorname{cosec} \theta}{\cot \theta} \\ & =2 \sec \theta=\mathrm{RHS} \end{aligned} \]