If cos²x + cos⁴x = 1, then tan²x + tan⁴x = ?

Solution (By Examveda Team)

$$\eqalign{ & {\cos ^2}x + {\cos ^4}x = 1 \cr & \Rightarrow {\cos ^4}x = 1 - {\cos ^2}x \cr & \Rightarrow {\cos ^4}x = {\sin ^2}x \cr & \Rightarrow {\cos ^2}x{\cos ^2}x = {\sin ^2}x \cr & \Rightarrow {\text{co}}{{\text{s}}^2}x = {\text{ta}}{{\text{n}}^2}x\,\,\,....(i) \cr & \Rightarrow {\cos ^4}x = {\text{ta}}{{\text{n}}^4}x\,\,\,....(ii) \cr & ta{n^2}x + {\tan ^4}x \cr & {\text{co}}{{\text{s}}^2}x + {\text{co}}{{\text{s}}^4}x = 1 \cr & {\text{ta}}{{\text{n}}^2}x + {\text{ta}}{{\text{n}}^4}x = 1 \cr & \left( {{\text{From eq}}{\text{. (i) and (ii)}}} \right) \cr} $$