cos(a+b) = cos(a)cos(b) -sin(a)sin(b)

cos 3t = cos(2t+t) = cos(2t)cos(t) - sin(2t)sin(t)

cos(2t) = cos^2(t)-sin^2(t)

sin(2t) = 2 sin(t) cos(t)

cos(2t)cos(t) - sin(2t)sin(t) = (cos^2(t)-sin^2(t))cos(t) - 2 sin(t) cos(t) sin(t)

= cos^3(t)-sin^2(t)cos(t) - 2 sin^2(t)cos(t)

= cos^3(t) - 3 sin^2(t)cos(t) (proved)

Do you know this identity?

cos(a + b) = cos(a).cos(b) - sin(a).sin(b) → suppose that: a = b = t

cos(t + t) = cos(t).cos(t) - sin(t).sin(t)

cos(2t) = cos²(t) - sin²(t) ← memorize this result as (1)

Do you know this identity?

sin(a + b) = sin(a).cos(b) + cos(a).sin(b) → suppose that: a = b = t

sin(t + t) = sin(t).cos(t) + cos(t).sin(t)

sin(2t) = 2.sin(t).cos(t) ← memorize this result as (2)

You know this identity?

cos(a + b) = cos(a).cos(b) - sin(a).sin(b) → suppose that: a = 2t and suppose that: b = t

cos(2t + t) = cos(2t).cos(t) - sin(2t).sin(t)

cos(3t) = cos(2t).cos(t) - sin(2t).sin(t) → recall (1): cos(2t) = cos²(t) - sin²(t)

cos(3t) = [cos²(t) - sin²(t)].cos(t) - sin(2t).sin(t) → recall (2) : sin(2t) = 2.sin(t).cos(t)

cos(3t) = cos³(t) - sin²(t).cos(t) - 2.sin²(t).cos(t)

cos(3t) = cos³(t) - 3.sin²(t).cos(t)

hint;

cos(3t)=cos(2t+t)=cos(2t)cos(t)-sin(2t)sin(t)