Question

How do you verify the identify `tantheta+cottheta=secthetacsctheta`?

Answer 1

see explanation.

Explanation:

We attempt to show by manipulation that the left side has the same form as the right side.

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

`color(orange)"Reminder"`

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

left side `=tantheta+cottheta`

`=(sintheta)/(costheta)+(costheta)/sintheta`

To combine these fractions we require a common denominator of `costhetasintheta.`

`rArr(sintheta)/(costheta)xx(sintheta)/(sintheta)+(costheta)/(sintheta)xx(costheta)/(costheta)`

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

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

`color(red)(bar(ul(|color(white)(a/a)color(black)(sectheta=1/(costheta)" and " csctheta=1/(sintheta))color(white)(a/a)|)))`

`rArr(sin^2theta+cos^2theta)/(costhetasintheta)=1/(costheta)xx1/(sintheta)`

`=secthetacsctheta=" right side"rArr" verified"`