Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\(\frac{(\text{1 + tan² A})}{(\text{1 + cot² A})} = (\frac{\text{1 - tan A }}{\text{ 1 - cot A}})^²= \text{tan² A}\)
\(\frac{(\text{1 + tan² A})}{(\text{1 + cot² A})} = (\frac{\text{1 - tan A }}{\text{ 1 - cot A}})^²= \text{tan² A}\)
Left Hand Side (LHS):\(\frac{(\text{1 + tan² A})}{(\text{1 + cot² A})} \)
\(= \frac{\text{sec² A}}{\text{cosec² A}}\)
\(= \frac{(\frac{1}{\text{cos}^2A})}{\text{(sin}^2A)}\)
\(= (\frac{1}{\text{cos² A}}) × (\frac{\text{sin² A}}{1})\)
\(=\text{ tan² A}\)
Right Hand Side (RHS)