The trigonometric identity `cos^4x+sin^2x*cos^2x + sin^2x + tan^2x = 1/(cos^2 x)` has to be proved.

Start with the left hand side:

`cos^4x+sin^2x*cos^2x + sin^2x + tan^2x`

= `cos^4x+sin^2x*cos^2x + sin^2x + (sin^2x)/(cos^2x)`

= `cos^4x+(1 - cos^2x)*cos^2x + 1 - cos^2x + (1 - cos^2x)/(cos^2x)`

= `cos^4x+cos^2x - cos^4x + 1 - cos^2x + (1 - cos^2x)/(cos^2x)`

= `1 + (1 - cos^2x)/(cos^2x)`

= `(cos^2x)/(cos^2x) + (1 - cos^2x)/(cos^2x)`

` `=> `1/(cos^2x)`

**This proves that** ` cos^4x+sin^2x*cos^2x + sin^2x + tan^2x =
1/(cos^2 x)`

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