If `cos alpha =1/2(x+1/x),cos beta=1/2(y+1/y)` then evaluate `cos(alpha-beta)`

Let alpha, beta be such that pi lt alpha lt beta lt 3pi, 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 alpha-beta lt3 pi . If sin alpha+ sin beta=(-21)/65 and cos alpha+ cos beta=(-27)/65 , then the value of cos((alpha-beta)/2) is

If sin alpha + sin beta = p "and" cos alpha + cos beta = q "then the value of " "tan" (alpha-beta)/2 will be-

If x cos alpha + y sin alpha = x cos beta + y sin beta "then" (2(1-cos (alpha - beta)))/(cos beta - cos alpha)^(2)=

If (sin (theta- alpha))/(sin (theta- beta))=(x)/(y) and (cos (theta- alpha))/(cos(theta-beta))=(a)/(b) , then show that cos(alpha- beta)=(ax+by)/(bx+ay) .

In sin alpha+ sin beta=(1)/(2) and cos alpha+ cos beta =(5)/(4) , find the value of tan alpha+tan beta .

If tan alpha =(1+2^(-x))^(-1) and tan beta =(1+2^(x+1))^(-1) then the value of (alpha + beta) is-

P( asec alpha, b tan alpha ) and Q (a sec beta, b tan beta) are two given points on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2)) = 1. Show that the equation of the chord PQ is (x)/(a) cos (alpha - beta)/(2) - (y)/(b) sin (alpha + beta)/(2) = cos (alpha+beta)/(2).

If x cos alpha + y sin alpha = x cos beta + y sin beta "then" "tan" (alpha + beta)/2=