Using mean value theorem, show that `(beta-alpha)/(1+beta^2) < tan^(-1)beta-tan^(-1)alpha < (beta-alpha)/ (1+alpha^2) , beta > alpha > 0.`

Prove , using mean value theorem, that |sin alpha -sin beta | le| alpha-beta|, alpha, beta in RR .

alpha beta x ^(alpha-1) e^(-beta x ^(alpha))

Find the value of x such that ((x+alpha)^2-(x+beta)^2)/(alpha+beta)=(sin (2theta))/(sin^2 theta) , where alpha and beta are the roots of the equation t^2-2t+2=0 .

If alpha, beta are the roots of x^(2) - px + q = 0 and alpha', beta' are the roots of x^(2) - p' x + q' = 0 , then the value of (alpha - alpha')^(2) + (beta + alpha')^(2) + (alpha - beta')^(2) + (beta - beta')^(2) is

Given alpha,beta, respectively, the fifth and the fourth non-real roots of units, then find the value of (1+alpha)(1+beta)(1+alpha^2)(1+beta^2)(1+alpha^4)(1+beta^4)

Find the values of the parameter a such that the rots alpha,beta of the equation 2x^2+6x+a=0 satisfy the inequality alpha//beta+beta//alpha<2.

If alpha and beta are the roots of the equation 3x^(2) - 5x + 2 = 0 , find the value of (i) (alpha)/(beta) + (beta)/(alpha) (ii) alpha-beta (iii) (alpha^(2))/(beta) + (beta^(2))/(alpha)

If alpha and beta be the roots of equation x^(2) + 3x + 1 = 0 then the value of ((alpha)/(1 + beta))^(2) + ((beta)/(1 + alpha))^(2) is equal to

If alpha and beta are acute such that alpha+beta and alpha-beta satisfy the equation tan^(2)theta-4tan theta+1=0 , then (alpha, beta ) =

Evaluate {:[( cos alpha cos beta , cos alpha sin beta , -sin alpha ),( -sin beta , cos beta, 0),( sin alpha cos beta, sin alpha sin beta, cos alpha ) ]:} =0