=> (1 +sec x) / (sec x)

=> ( 1+ (1/cos x)) / (1/cosx)

musltply each term by cos x

=> (cos x + 1) / 1

multiply both numerator and the denominator by (1- cos x)

=> (1 + cos x)(1- cos x) / (1-cos x)

=> (1 - cos^2x)...

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=> (1 +sec x) / (sec x)

=> ( 1+ (1/cos x)) / (1/cosx)

musltply each term by cos x

=> (cos x + 1) / 1

multiply both numerator and the denominator by (1- cos x)

=> (1 + cos x)(1- cos x) / (1-cos x)

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

as cos ^2x + sin ^2x = 1,

1 - cos ^2 x = sin^2 x

substituting this,

=>sin^2x /( 1- cosx)

**hence (1 +sec x) / (sec x) = sin^2x /( 1- cosx)**

The identity `(1 + sec x)/sec x = (sin^2 x)/(1 - cos x)` has to be proved.

`(sin^2 x)/(1 - cos x)`

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

=> `((1 - cos x)(1 + cos x))/(1 - cos x)`

=> `(1 + cos x)`

=> `1 + 1/sec x`

=> `(sec x + 1)/sec x`

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