Question

How do you verify the identify `tan^2theta+1=sec^2theta`?

Answer 1

see explanation

Explanation:

The following `color(blue)"trigonometric identies"` are 'useful'

`color(orange)"Reminders"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(tantheta=(sintheta)/(costheta))color(white)(a/a)|)))`

`color(red)(|bar(ul(color(white)(a/a)color(black)(sin^2theta+cos^2theta=1)color(white)(a/a)|)))" and " color(red)(|bar(ul(color(white)(a/a)color(black)(sectheta=1/costheta)color(white)(a/a)|)))`

the left side `=tan^2theta+1=(sin^2theta)/(cos^2theta)+(cos^2theta)/(cos^2theta)`

Expressing as a single fraction gives.

`(sin^2theta+cos^2theta)/(cos^2theta)=1/(cos^2theta)=sec^2theta="right side hence verified"`