Question

Eliminate x from equations sin x + cos x = a and sin^3 + cos^3 x = b

Answer 1

sinx + cosx = a ------(1)
take square both sides,

sin²x + cos²x + 2sinx.cosx = a²
[ use, sin²x + cos²x = 1 ]

1 + 2sinx.cosx = a²
sinx.cosx = (a² -1)/2 -------(2)

again,
sin³x + cos³x = b
(sinx + cosx)³ -3sinx.cosx(sinx+ cosx) = b
[put eqns (1) and (2) ]

a³ - 3(a² -1)/2(a ) = b

2a³ -3a(a²-1) = 2b

2a³ -3a³ +3a = 2b

-a³ +3a = 2b

a³-3a +2b = 0 ( answer )