Prove that cos theta /1-sin theta = 1+sin theta / cos theta ?

2 Answers
Jun 19, 2018

See below

Explanation:

If I understood corretcly (please format your questions!!!!), you want to prove that

#cos(theta)/(1-sin(theta)) = (1+sin(theta))/cos(theta)#

This can be proven by cross-multiplication: multiply both sides by #cos(theta)(1-sin(theta))#, i.e. by both denominators to get

#cos^2(theta) = (1+sin(theta))(1-sin(theta))#

On the right hand side we have the expression #(a+b)(a-b)=a^2-b^2#, so the expression becomes

#cos^2(theta) = 1-sin^2(theta)#

Which is true, because it derives immediately from the fundamental trigonometric equation

#cos^2(theta)+sin^2(theta)=1#

Jun 19, 2018

#LHS=cos(theta)/(1-sin(theta)) #

#=cos^2(theta)/(costheta(1-sin(theta)) #

#=(1-sin^2(theta))/(costheta(1-sin(theta)) #

#=((1-sin(theta))(1+sintheta))/(costheta(1-sin(theta)) #

#= (1+sin(theta))/cos(theta)=RHS#