trig identity?

Question

sin(θ-45º) = √2 / 2 (sinθ-cosθ)


tan(A + π/4) = tanA+1/1-tanA


2+cos²A - 3cos^4A = sin²A(2+3cos²A)

Answer 1

1.
Use the difference of two angle identity:

sin (θ-45º) = sin θ cos 45 - cosθ sin 45

= sqrt2 / 2 sin θ - sqrt2/2 cos θ

= sqrt2 / 2 (sinθ - cosθ)

Answer 2

sin(A - 45) =
sinAcos(45) - sin(45)cos(A) =
sin(A)(sqrt(2)/2) - (sqrt(2)/2)cosA =
(sqrt(2)/2)(sinA - cosA)

so

sin(A - 45) = (sqrt(2)/2)(sin(A) - cos(A))

----------------------------------------------------------

tan(A + (pi/4)) =
(tanA + tan(pi/4))/(1 - tan(A)tan(pi/4)) =
(tan(A) + 1)/(1 - tanA)

so

tan(A + (pi/4)) = (tan(A) + 1)/(1 - tan(A))

-----------------------------------

2 + cos(A)^2 - 3cos(A)^4 =
(2 + 3cos(A)^2)(1 - cos(A)^2) =
(2 + 3cos(A)^2)(sin(A)^2)

so

2 + cos(A)^2 - 3cos(A)^4 = (sin(A)^2)(2 + 3cos(A)^2)

By the way, for the ones that contained the theta symbol, i just used A, because i was more familiar with using that one to stand for Theta.