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

Using Lagranges mean value theorem, show that sinx 0.

(a) show that alpha^2 + beta^2 + alpha beta =0

In [0, 1] Lagrange's mean value theorem is not applicable to

lf cos^2 alpha -sin^2 alpha = tan^2 beta , then show that tan^2 alpha = cos^2 beta-sin^2 beta .

If alpha and beta are the complex cube roots of unity, show that alpha^4+beta^4 + alpha^-1 beta^-1 = 0.

If alpha and beta are the roots of the equation x^(2)+alpha x+beta=0 such that alpha ne beta , then the number of integral values of x satisfying ||x-beta|-alpha|lt1 is

The lengths of the sides of a triangle are alpha-beta, alpha+beta and sqrt(3alpha^2+beta^2), (alpha>beta>0) . Its largest angle is

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

If alpha , beta are the roots of x^2 +x+1=0 then alpha beta + beta alpha =

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)