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You need to remember the following trigonometric identity such that:

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

Hence, substituting `1/(cos^2 x)`  for `1 + tan^2 x`  yields:

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

Since the definition of secant function is the following `sec x = 1/cos x` , hence, `sec^2 x = (1/ cos x)^2`

Substituting `(1/ cos x)^2`  for `sec^2 x`  yields:

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

Since using the trigonometric identities `1 + tan^2 x = 1/(cos^2 x)`  and `sec x = 1/cos x` , yields the left side equal to the right side, hence, the given identity `1 + tan^2 x = sec^2 x`  holds.

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Use the identities `tanx = sinx/cosx`  and `secx = 1/cosx`

Then `tan^2x +1 = (sinx/cosx)^2 + 1= (sin^2x)/(cos^2x) + (cos^2x)/(cos^2x)`

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

(using `sin^2x + cos^2x = 1`)

The proof is complete

 

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