# Prove that : 2 tan^−1(√(a−b/a+b) tan(x2))=cos^−1(acosx+ba+bcosx) - Mathematics

Sum

Prove that :

2 tan^-1 (sqrt((a-b)/(a+b))tan(x/2))=cos^-1 ((a cos x+b)/(a+b cosx))

#### Solution

2 tan^-1 (sqrt((a-b)/(a+b))tan(x/2))

=cos^(-1){(1-(sqrt((a-b)/(a+b))tan(x/2))^2)/(1+(sqrt((a-b)/(a+b))tan(x/2))^2)} [∵ 2 tan^(-1) (x)=cos^(−1)((1−x^2)/(1+x^2))]

=cos^(-1) {(1-(a-b)/(a+b)tan^2(x/2))/(1+(a-b)/(a+b)tan^2(x/2))}

=cos^(-1){(a+b-(a-b)tan^2(x/2))/(a+b+(a-b)tan^2(x/2))}

=cos^(-1){(a+b-atan^2(x/2)+btan^(x/2))/(a+b+atan^2(x/2)-btan^(x/2))}

=cos^(-1) {(a(1-tan^2(x/2))+b(1+tan^2(x/2)))/(a(1+tan^2(x/2))+b(1-tan^2(x/2)))}

=cos^(-1) {(a((1-tan^2(x/2))/(1+tan^2(x/2)))+b((1+tan^2(x/2))/(1+tan^2(x/2))))/(a((1+tan^2(x/2))/(1+tan^2(x/2)))+b((1-tan^2(x/2))/(1+tan^2(x/2))))}

=cos^(-1){(a((1-tan^2(x/2))/(1+tan^2(x/2)))+b)/(a+b((1-tan^2(x/2))/(1+tan^2(x/2))))}

=cos^(-1){(acosx+b)/(a+bcosx)}

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Is there an error in this question or solution?
2014-2015 (March) Patna Set 2

Share