text: # Verify the following identity: `tanx(cot x + tan x) = sec^2x`
  
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  We have to verify that `tanx(cot x + tan x) = sec^2x`
  
  => `tan x*cot x+ tan^2x`
  
  => `1 + tan^2x`
  
  => `1 + (sin^2x)/(cos^2x)`
  
  => `(cos^2x + sin^2x)/(cos^2x)`
  
  =` `> `1/(cos^2x)`
  
  => `sec^2x`
  
  This proves that `tan x(cot x + tan x) = sec^2 x`
  
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  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`
  
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  L:H:S ≡ tanx(cotx + tanx)
  
  = tanx.cotx + tan²x
  
  ⇒ tanθ.cotθ = 1
  
  = 1 + tan²x
  
  ⇒ 1 + tan²θ = sec²θ
  
  = sec²x
  
  = R:H:S