Prove that : `(a^(-1))/(a^(-1)+b^(-1))+(a^(-1))/(a^(-1)-b^(-1))=(2b^2)/(b^2-a^2)`

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

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

If a, b, c are in geometric progression, then prove that : (1)/(a^(2)-b^(2))+(1)/(b^(2))=(1)/(b^(2)-c^(2))

The asymptotes of the hyperbola (x^(2))/(a_(1)^(2))-(y^(2))/(b_(1)^(2))=1 and (x^(2))/(a_(2)^(2))-(y^(2))/(b_(2)^(2))=1 are perpendicular to each other. Then, (a) a_(1)/a_(2)=b_(1)/b_(2) (b) a_(1)a_(2)=b_(1)b_(2) (c) a_(1)a_(2)+b_(1)b_(2)=0 (d) a_(1)-a_(2)=b_(1)-b_(2)

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

Using properties of determinants, prove that: |(1+a^(2)-b^(2),2ab,-2b),(2ab,1-a^(2)+b^(2),2a),(2b,-2a,1-a^(2)-b^(2))|=(1+a^(2)+b^(2))^(3)

If a, b and c are in G.P. then prove that 1 a 2 - b 2 + 1 b 2 = 1 b 2 - c 2 . 1/(a^2-b^2)+1/(b^2)=1/(b^2-c^2)dot

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

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

Prove that: |1a a^2-b c1bb^2-c a1cc^2-a b|=0