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

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Answer 1 · 2018-02-23T16:22:05

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Answer 2 · 2018-02-23T19:22:55

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)#

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