Prove that `tan(ilog_e((a-ib)/(a+ib)))=(2ab)/(a^2-b^2)` (where `a, b in R^+`)

If alpha and beta are the two different roots of equations alpha cos theta+b sin theta=c , prove that (a) tan (alpha-beta)=(2ab)/(a^(2)-b^(2)) (b) cos(alpha+beta)=(a^(2)-b^(2))/(a^(2)+b^(2))

Prove log ( a^(2))/(ba) + log((b^(2))/(ab))=0

Prove that matrix [((b^(2)-a^(2))/(a^(2)+b^(2)),(-2ab)/(a^(2)+b^(2))),((-2ab)/(a^(2)+b^(2)),(a^(2)-b^(2))/(a^(2)+b^(2)))] is orthogonal.

Prove that tan(pi/4+1/2cos^(-1)(a/b))+tan(pi/4-1/2cos^(-1)(a/b))=(2b)/a .

((1 - i)/(1+ i))^(100) = a + ib then (a,b) is ……….

If sinalpha+sinbeta=a and cosalpha+cosbeta=b , prove that tan((alpha-beta)/2)=+-sqrt((4-a^2-b^2)/(a^2+b^2)) .

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)))

Prove that |(1,a,a^(2)-bc),(1,b,b^(2)-ca),(1,c,c^(2)-ab)|= 0 .

Prove that (tan^(-1)1/e)^2+(2e)/((e^2+1)<(tan^(-1)e)^2+2/(sqrt(e^2+1))