Express as a product: 1 + cos(a)

Answer 1

1 + cosa = cos0 + cosa =

sin^2(a) + cos^2(a) + cosa

let x = cosa, then =

x^2 + x +sin^2(a)

x= -.5 + or - .5sqr(1 -4sin^2(a))

then corresponding factors are

(x+5 -.5(sqr(1-4sin^2(a))(x+5 +.5(sqr(1-4sin^2(a))

replace x with cosa

= (cosa + .5 -.5(sqr(1-4sin^2(a))(cosa + .5 -.5(sqr(1-4sin^2(a))

or= (cosa + 1/2 - (1/2)sqr((1-2sina)(1+2sina))(cosa + 1/2 + (1/2)sqr((1-2sina)(1+2sina))

or write 1+cosa as product of differences and sum (a-b)(a+b) = a^2 -b^2 or (a-bi)(a+bi) = a^2 +b^2 with a=1 and b = isqrcosa

= (1-isqrcosa)(1+isqrcosa) where i makes these imaginary factors

Answer 2

1 + cos(a) = cos0 + cos(a) = 2cos(a/2)·cos(a/2) = 2cos²(a/2)