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

2 Answers
R. Nguyen
Apr 6, 2018

(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)

Aviv S.
Apr 6, 2018

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!

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