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

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Sum
 

Prove that :

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

 
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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)
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