Solve : `sin^(-1)((2alpha)/(1+alpha^2)) + sin^(-1)( (2beta)/(1+beta^2)) =2 tan^(-1) x, |alpha|le1, |beta|le1`

x: (sin ^ (- 1) (2 alpha)) / (1 + a ^ (2)) + (sin ^ (- 1) (2 beta)) / (1 + beta ^ (2)) = 2tan ^ (-1) x

Show taht 2tan ^ (- 1) (tan ((alpha) / (2)) tan ((pi) / (4) - (beta) / (2))) = tan ^ (- 1) ((sin alpha cos beta) / (cos alpha + sin beta))

If tan^(-1)(alpha+i beta)=a+ib, then a= 1). (1)/(2)tan^(-1)((2 alpha)/(1-alpha^(2)-beta^(2))) 2) . (1)/(2)tan^(-1)((2 alpha)/(1+alpha^(2)+beta^(2))) 3). tan^(-1)((2 alpha)/(1-alpha^(2)-beta^(2))) 4). (1)/(2)tan^(-1)((2 beta)/(1+alpha^(2)+beta^(2)))

If alpha le sin^(-1) x + cos^(-1) x - tan^(-1) x le beta , then

If tan alpha=(1+2^(-x))^(-1), tan beta=(1+2^(x+1))^(-1) then alpha+beta equals

if alpha,beta be the roots of the equation ax^(2)+bx+c=0 then the roots of a(2x+1)^(2)+b(2x+1)(x-1)+c(x-1)^(2)=0 are (i) (2 alpha+1)/(alpha-1),(2 beta+1)/(beta-1) (ii) (2 alpha-1)/(alpha+1),(2 beta-1)/(beta+1) (iii) (alpha+1)/(alpha-2),(beta+1)/(beta-2)( iv) (2 alpha+3)/(alpha-1),(2 beta+3)/(beta-1)

tan alpha=(1+2^(-x))^(-1),tan beta=(1+2^(x+1))^(-1)alpha+beta

If sin (alpha + beta)=1, sin (alpha- beta)= (1)/(2) , then tan (alpha + 2beta) tan (2alpha + beta) =

Prove that: (alpha^(3))/(2)csc^(2)((1)/(2)tan^(-1)(alpha)/(beta))+(beta^(2))/(2)sec^(2)((1)/(2)tan^(-1)(beta)/(alpha))=(alpha+beta)(a^(2)+beta^(2))

If alpha and beta are the roots of the equation x^(2)-4x+1=0(alpha>beta) then find the value of f(alpha,beta)=(beta^(3))/(2)csc^(2)((1)/(2)tan^(-1)((beta)/(alpha))+(alpha^(3))/(2)sec^(2)((1)/(2)tan^(-1)((alpha)/(beta)))