Question

Question #246a9

Answer 1

see below

Explanation:

Left Hand Side:

`color(red)(1 + tan^2theta/(sec theta+1)=color(blue)(1+(sin^2theta/cos^2theta)/(1/costheta+1)`

`color(blue)(=1+(sin^2theta/cos^2theta)/((1+costheta)/cos theta)`

`color(blue)(=1+sin^2theta/cos^2theta*(cos theta)/(1+costheta)`

`color(blue)(=1+sin^2theta/cos^cancel2theta*cancel(cos theta)/(1+costheta)`

`color(blue)(=1+sin^2theta/(costheta(1+costheta)`

`color(blue)(=(cos theta+cos^2 theta+sin^2theta)/(costheta(1+cos theta))`

`color(blue)(=(cos theta+1)/(cos theta(1+cos theta)`

`color(blue)(=(1+cos theta)/(cos theta(1+cos theta)`

`color(blue)(=cancel(1+cos theta)/(cos theta cancel((1+cos theta)`

`color(blue)(=1/cos theta`

`color(blue)( :. = sec theta`

Answer 2

`LHS= 1 + tan^2theta/(sec theta+1)`

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

`= 1 + ((cancel(sectheta+1))(sectheta-1))/(cancel(sec theta+1))`

`=1+sectheta-1=sectheta=RHS`