Question

How do you simplify `3(tan^2 theta - sec^2 theta)`?

Answer 1

`3(tan^2(theta)-sec^2(theta))=- 3`

Explanation:

`tan(theta) = (sin(theta))/(cos(theta))`
`color(white)("XXXX")` `color(white)("XXXX")` `rarr tan^2(theta) = sin^2(theta)/cos^2(theta)`

`sec(theta) = 1/cos(theta)`
`color(white)("XXXX")` `color(white)("XXXX")` `sec^2(theta)=1/cos^2(theta)`

`tan^2(theta) - sec^2(theta)`
`color(white)("XXXX")` `color(white)("XXXX")` `=sin^2(theta)/cos^2(theta) - 1/cos^2(theta)`

`color(white)("XXXX")` `color(white)("XXXX")` `=(sin^2(theta)-1)/(cos^2(theta)`

and since `sin^2(theta)+cos^2(theta) = 1`
`color(white)("XXXX")` `color(white)("XXXX")` `= (-cos^2(theta))/cos^2(theta)`

`color(white)("XXXX")` `color(white)("XXXX")` `= - 1`

`3(tan^2(theta)-sec^2(theta))`
`color(white)("XXXX")` `color(white)("XXXX")` `=- 3`

Answer 2

You can use the identity:

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

`=> 3(tan^2theta - (1+tan^2theta))`

`= 3(tan^2theta - 1 - tan^2theta)`

`= 3(-1) = color(blue)(-3)`