Question

How do you simplify `(sec^2x-1)/ sec^2x`?

Answer 1

Simplify `(sec^2 - 1)/sec^2`

Ans: sin^2 x

Explanation:

Replace `sec^2 = 1/(cos^2 x` into the expression , we get:
`(1/(cos^2 x - 1))/(1/(cos^2 x)) = ((1 - cos^2 x)/cos^2 x)((cos^2 x)/1) =`

Since`(1 - cos^2 x) = sin^2x`, therefore:

`(sec^2 x - 1)/sec^2 x = sin^2 x`

Answer 2

`sin^2x`

Explanation:

`1 + tan^2x=sec^2x`

so `" "sec^2x-1 = tan^2x`

Giving `" " (tan^2x)/(sec^2x)`

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

Giving `" " (tan^2x)(cos^2x)`

But `tan^2x =(sin^2x)/(cos^2x)`

Giving `" " (sin^2x) (cos^2x)/(cos^2x) = sin^2x`

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Clarification: By Tony B

Given:`" " (sec^2x-1)/sec^2x`

But `Sec^2x = 1 +tan^2x`

Giving:`" "(1+tan^2x-1)/(sec^2x)" "=" " tan^2x/sec^2x`

But `" "tanx=sinx/cosx" and "secx= 1/cosx`

`" "sin^2x/(cancel(cos^2x)) xxcancel(cos^2x)" "=" "sin^2x`

Answer 3

`=sin^2x`

Explanation:

`(sec^2x-1)/sec^2x`
`=cancel(sec^2x)/cancel(sec^2x)-1/sec^2x`
`=1-cos^2x`
`=sin^2x`