Question

Verify the identity of `sec^6X(secXtanX)-sec^4X(secXtanX)=sec^5Xtan^3X ?`

Answer 1

See below

Explanation:

`sec^6x(secxtanx)-sec^4x(secxtanx)=sec^4xsecxtanx(sec^2x-1)=sec^5xtanxtan^2x=sec^5xtan^3x`

Answer 2

`sec^6X(secXtanX)-sec^4X(secXtanX)=sec^5Xtan^3X`

Taking the `LHS`,

`sec^6X(secXtanX)-sec^4X(secXtanX)`

`=>(sec^6X-sec^4X)(secXtanX)`

`=>sec^4X(sec^2X-1)(secXtanX)`

`=>sec^4X(tan^2X)(secXtanX)` `color(white)(ww` `["as " color(red)(sec^2X-1 =tan^2X) ]`

Getting the `tanX` and `secX` together,

`=>sec^5Xtan^3X = RHS`

Hence Verified ! :)

Answer 3

Please see below.

Explanation:

We know that,

`color(red)((1)sec^2theta-1=tan^2theta`

Here,

`sec^6X(secXtanX)-sec^4X(secXtanX)=sec^5Xtan^3X`

We take,

`LHS=sec^6X(color(blue)(secXtanX))-sec^4X(color(blue)(secXtanX))`

`=color(blue)(secXtanX)(sec^6X-sec^4X)`

`=color(blue)(secXtanX)sec^4X(sec^2X-1)`

`=sec^5XtanX(color(red)(sec^2X-1))...toApply(1)`

`=sec^5XtanX(color(red)(tan^2X))`

`=sec^5Xtan^3X`

`=RHS`