If tan(A/2)=sqrt((1-x)/(1+x))*tan((theta)/2) then show that costheta=(cosA-x)/(1-x*cosA)?

2 Answers
Feb 23, 2018

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Feb 23, 2018

Given

tan(A/2)=sqrt((1-x)/(1+x))tan(theta/2)

=>tan(theta/2)=tan(A/2)*sqrt((1+x)/(1-x))

Now costheta

=(1-tan^2(theta/2))/(1+tan^2(theta/2))
Inserting the value of tan(theta/2)

=(1-tan^2(A/2)((1+x)/(1-x)))/(1+tan^2(A/2)((1+x)/(1-x))

=((1-x)-tan^2(A/2)(1+x))/((1-x)+tan^2(A/2)((1+x))

=((1-tan^2(A/2))-x(1+tan^2(A/2)))/((1+tan^2(A/2))-x(1-tan^2(A/2))

=((1-tan^2(A/2))/(1+tan^2(A/2))-x)/(1-x((1-tan^2(A/2))/(1+tan^2(A/2)))

=(cosA-x)/(1-xcosA)