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

Prove that 2tan^(-1)sqrt((b)/(a))=cos^(-1)((a-b)/(a+b))

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

prove that 2tan^(-1)(sqrt((a-b)/(a+b))tan((theta)/(2)))=cos^(-1)((a cos theta+b)/(a+b cos theta))

Tangents are drawn to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1,(a > b), and the circle x^2+y^2=a^2 at the points where a common ordinate cuts them (on the same side of the x-axis). Then the greatest acute angle between these tangents is given by (A) tan^(-1)((a-b)/(2sqrt(a b))) (B) tan^(-1)((a+b)/(2sqrt(a b))) (C) tan^(-1)((2a b)/(sqrt(a-b))) (D) tan^(-1)((2a b)/(sqrt(a+b)))

Let a, b and c be positive real numbers. Then prove that tan^(-1) sqrt((a(a + b + c))/(bc)) + tan^(-1) sqrt((b (a + b + c))/(ca)) + tan^(-1) sqrt((c(a + b+ c))/(ab)) = pi

If cos theta = (a cos phi + b) / (a + b cos phi) then (tan ((theta) / (2))) / (tan ((phi) / (2))) = (A) ( ab) / (a + b) (B) (a + b) / (ab) (C) sqrt ((a + b) / (ab)) (D) sqrt ((ab) / (a + b))

Prove that : cos ^(-1) ((1- a^(2))/(1+a)) + cos ^(-1)((1-b^(2))/(1+b^(2))) = 2 tan ^(-1) .(a+b)/(1-ab)