Question

How do you simplify `(1- tan^2θ) /( 1+ tan^2 θ)`?

Answer 1

`2cos^2theta-1` or `cos2theta`

Explanation:

Using the identities:
`1+tan^2theta= sec^2theta`
`1/sectheta= costheta`
`tantheta= sintheta/costheta`
`sin^2theta= 1-cos^2theta`
`2cos^2theta-1= cos2theta`

Start:
`(1-tan^2theta)/(1+tan^2theta)=`

`(1-tan^2theta)/(sec^2theta)=`

Split the numerator:

`1/sec^2theta-tan^2theta/sec^2theta=`

`cos^2theta-sin^2theta/cancel(cos^2theta)*cancel(cos^2theta)=`

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

`2cos^2theta-1=`

`cos2theta`

Answer 2

`cos2theta`

Explanation:

`"using the "color(blue)"trigonometric identities"`

`•color(white)(x)tantheta=sintheta/costheta`

`•color(white)(x)sin^2theta+cos^2theta=1`

`•color(white)(x)cos^2theta-sin^2theta=cos2theta`

`rArr(1-tan^2theta)/(1+tan^2theta)`

`=(1-sin^2theta/cos^2theta)/(1+sin^2theta/cos^2theta)xxcos^2theta/cos^2theta`

`=(cos^2theta-sin^2theta)/(cos^2theta+sin^2theta)`

`=cos^2theta-sin^2theta=cos2theta`