Question

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

Answer 1

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

Explanation:

First, split the fraction.

`(2 - sec^2(x))/sec^2(x)`

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

Because `cos(x) = 1/sec(x)`, `color(red)(cos^2(x) = 1/sec^2(x))`

`= 2/color(red)(sec^2(x)) - sec^2(x)/sec^2(x)`

`= color(blue)(2cos^2(x) - 1)`

Observe the following:

`color(blue)cos(2x)`

`= cos^2(x) - sin^2(x)`

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

`= color(blue)(2cos^2(x) - 1)`

Therefore,

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

Answer 2

It simplifies to `cos(2x)`.

Explanation:

Use these identites:

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

`secx=1/cosxqquadcolor(blue)=>qquadsec^2x=1/cos^2x`

First, split the fraction:

`color(white)=(2-sec^2x)/sec^2x`

`=2/sec^2x-sec^2x/sec^2x`

`=2/sec^2x-1`

`=2*1/sec^2x-1`

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

`=2*cos^2x-1`

`=2cos^2x-1`

`=cos(2x)`

Hope this helped!