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

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

Show that cos^(-1) ((cos alpha+cos beta)/(1+cosalpha cosbeta))=2 tan^(-1)(tan (alpha/2) tan (beta/2))

If the eccentric angles of the extremities of a focal chord of an ellipse x^2/a^2 + y^2/b^2 = 1 are alpha and beta , then (A) e = (cos alpha + cos beta)/(cos (alpha + beta)) (B) e= (sin alpha + sin beta)/(sin(alpha + beta)) (C) cos((alpha-beta)/(2)) = e cos ((alpha + beta)/(2)) (D) tan alpha/2.tan beta/2 = (e-1)/(e+1)

Prove that: (alpha^3)/2cos e c^2(1/2tan^(-1)(alpha/beta))+(beta^3)/2s e c^2(1/2tan^(-1)(beta/alpha))=(alpha+beta)(alpha^2+beta^2) .

Prove that: (alpha^3)/2cos e c^2(1/2tan^(-1)alpha/beta)+(beta^3)/2sec^2(1/2tan^(-1)beta/alpha)=(alpha+beta)(a^2+beta^2)dot

If alpha and beta are zeroes of a quadratic polynomial ax^(2)+bx+c . Find the value of: (i) alpha^(2)-beta^(2) (ii) alpha^(3)+beta^(3) (iii) alpha^(4)beta+beta^(4)alpha (iv) sqrt((alpha)/(beta))+sqrt((beta)/(alpha)) (v) (alpha^(2))/(beta)+(beta^(2))/(alpha) (alpha^(-1)+(1)/(alpha^(-1)))(beta^(-1)+(1)/(beta^(-1)))

If alpha&beta are the roots of the quadratic equation a x^2+b x+c=0 , then the quadratic equation, a x^2-b x(x-1)+c(x-1)^2=0 has roots: alpha/(1-alpha),beta/(1-beta) (b) alpha-1,beta-1 alpha/(alpha+1),beta/(beta+1) (d) (1-alpha)/alpha,(1-beta)/beta

If alpha,beta re the roots of a x^2+c=b x , then the equation (a+c y)^2=b^2y in y has the roots a. alphabeta^(-1),alpha^(-1)beta b. alpha^(-2),beta^(-2) c. alpha^(-1),beta^(-1) d. alpha^2,beta^2

If alpha,beta(alpha < beta) are the roots of equation 6x^2+11="" x+3="0 , then which following real? (a) cos^(-1)alpha (b) sin^(-1)beta (c) cosec^(-1)alpha (d) both cot^(-1)alpha and cot^(-1)beta

If alpha le sin^(-1)x+cos^(-1)x +tan^(-1)xlebeta then find the value of alpha and beta