An alternative method is to take `sec^2 x ` and replace it by `1/(cos^2 x).`
Replace `1/(cos^2 x)` by `1+ tan^2 x` (the basic formula of trigonometry `1+tan^2 x = 1/(cos^2 x)` ).
Use this substitution you've made in the identity to be proven.
`tan x(cot x + tan x) = 1+tan^2 x`
Opening the brackets, you'll have`tanx*cotx + tan^2x = 1+tan^ 2 x`
The cotangent function is the inverse of tangent, therefore tanx*cot x = 1.
`1+ tan^2x = 1+tan^ 2 x`
ANSWER: The last line proves the identity`tan x(cot x + tan x) = sec^2 x`