Answer:
Explanation:
we know : all reals x : (cosx)² + (sinx)² = 1
so : (cosx)² = 1 - (sinx)²
put the value of (cosx)² in equation : (cosx)²+sinx +1 =0
you have : 1 - (sinx)² + sinx +1 = 0
multiply by -1 : (sinx)² - sinx -2 = 0
let : sinx = t ........ Without forgetting -1 ≤ t ≤ 1
t² -t -2 = 0 quadrtic equation when : a = 1 b = -1 c = -2
Δ = b² - 4ac Δ = (-1)² -4(1)(-2) = 9 = 3²
t1 = (- b +√Δ)/2a = (1+3)/2 = 2 refused
t2 = (- b -√Δ)/2a = (1-3)/2 =-1 accept because : -1 ≤ t ≤ 1
but : sinx = t so : sinx = - 1
we know sin a = sinb equi : a = b+2kπ or a = π - b +2kπ k in Z......(*)
look : -1 = - sin(π/2) = sin(- π/2) .... because sin(-c) = - sin(c)
so : sinx = sin(- π/2)
by (*) : x = - π/2 + 2kπ or x =π -(- π/2) + 2kπ
conclusion : x = - π/2 + 2kπ or x =3π/2 + 2kπ .....k in Z