Question

(a) tan2 A
(b) cot2 A
(c) sec2 A
(d) cosec2 A

Answer 1

(a) tan² A
$$\begin{gathered} \frac{1+{\tan }^{2}A}{1+{\cot }^{2}A} \\ \text{=}\frac{1+\frac{{\text{sin}}^{2}A}{{\text{cos}}^{2}A}}{1+\frac{{\text{cos}}^{2}A}{{\text{sin}}^{2}A}} \\ \text{=}\frac{\frac{{\text{cos}}^{2}A+s{\text{in}}^{2}A}{{\text{cos}}^{2}A}}{\frac{{\text{sin}}^{2}A+c{\text{os}}^{2}A}{{\text{sin}}^{2}A}} \\ \text{=}\frac{\frac{1}{{\text{cos}}^{2}A}}{\frac{1}{{\text{sin}}^{2}A}}(\because c{\text{os}}^{2}A+s{\text{in}}^{2}A=1) \\ \text{=}\frac{{\text{sin}}^{2}A}{{\text{cos}}^{2}A}={\text{tan}}^{2}A \end{gathered}$$