Let `alpha, beta ` be such that it ` pi < alpha - beta <3 pi`. if `sin alpha + sin beta =-21/65 ` and `cos alpha+cos beta = -27 / 65`then the value of `cos ((alpha-beta)/2) ` is

Let alpha, beta be such that pi lt alph-betalt3piif sin alpha+sinbeta=-21/65and cos alpha+cos beta =-27/65, then the value of cos""(alpha-beta)/(2), is

Let alpha, beta be such that pi lt alpha -beta lt 3 pi. If sin alpha + sin beta =- (21)/(65) and cos alpha + cos beta =- (27)/(65) and if the value of cos "" (alpha - beta)/(2) =(-a)/(sqrtb), then a xx b=

Let alpha,beta be real numbers such that all alpha,beta,alpha+beta,alpha-beta cannot be odd multiple of pi and 3(cos alpha+cos beta)+1+cos(alpha-beta)=sin alpha*sin beta then for those alpha,beta in R for which tan((alpha)/(a)),,tan((beta)/(2)) are real,which of the following is(are) true?

Let f:[0,2] to RR be the function defined by f(x)=(3-sin (2pix)) sin (pi x-(pi)/(4)) - sin (3pi x+(pi)/(4)) If alpha,beta in [0,2] are such that {x in [0,2]: f(x) ge0}= [alpha, beta] , then the value of beta- alpha is._________

If alpha and beta are angles such that alpha+beta=(2 pi)/(3) then number of integers in range of (sin alpha sin beta+4cos alpha cos beta)

Let A,B be integers and let alpha,beta be the roots of the equation,x^(2)-x-1=0, where alpha!=beta For n=0,1,2,..., Let a_(n)=A alpha^(-n)+B beta^(-n)

Let alpha and beta be two distinct complex numbers, such that abs(alpha)=abs(beta) . If real part of alpha is positive and imaginary part of beta is negative, then the complex number (alpha+beta)//(alpha-beta) may be