How do you verify the identify tantheta+cottheta=secthetacsctheta?
1 Answer
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"
