How do you prove the identity sec u - tan u = cos u/(1+sin u)secu−tanu=cosu1+sinu ?
1 Answer
Feb 22, 2017
See explanation...
Explanation:
Use:
sec u = 1/cos usecu=1cosu
tan u = sin u/cos utanu=sinucosu
cos^2 u + sin^2 u = 1cos2u+sin2u=1
Then:
sec u - tan u = 1/cos u - sin u/cos usecu−tanu=1cosu−sinucosu
color(white)(sec u - tan u) = (1-sin u)/cos usecu−tanu=1−sinucosu
color(white)(sec u - tan u) = (1-sin u)/cos u*(1+sin u)/(1+sin u)secu−tanu=1−sinucosu⋅1+sinu1+sinu
color(white)(sec u - tan u) = (1-sin^2 u)/(cos u(1+sin u))secu−tanu=1−sin2ucosu(1+sinu)
color(white)(sec u - tan u) = cos^2 u/(cos u(1+sin u))secu−tanu=cos2ucosu(1+sinu)
color(white)(sec u - tan u) = cos u/(1+sin u)secu−tanu=cosu1+sinu