Rewrite with only cosx and sinx: sin3x - cosx?

Trigonometric identity: sin(3x) = 3 sin(x) - 4 sin³(x)

Hence, sin(3x) - cos(x) = 3 sin(x) - 4 sin³(x) - cos(x)

Do you know this identity?

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

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

cos(2x) = cos²(x) - sin²(x)

cos(2x) = cos²(x) - [1 - cos²(x)]

cos(2x) = cos²(x) - 1 + cos²(x)

cos(2x) = 2.cos²(x) - 1 ← 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 = x

sin(x + x) = sin(x).cos(x) + cos(x).sin(x)

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


You know this identity

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

sin(2x + x) = sin(2x).cos(x) + cos(2x).sin(x)

sin(3x) = sin(2x).cos(x) + cos(2x).sin(x) → recall (2)

sin(3x) = [2.sin(x).cos(x)].cos(x) + cos(2x).sin(x) → recall (1)

sin(3x) = [2.sin(x).cos(x)].cos(x) + [2.cos²(x) - 1].sin(x)

sin(3x) = 2.sin(x).cos²(x) + 2.cos²(x).sin(x) - sin(x)

sin(3x) = 3.cos²(x).sin(x) - sin³(x)

sin(3x) = 3.[1 - sin²(x)].sin(x) - sin³(x)

sin(3x) = 3.sin(x) - 3.sin³(x) - sin³(x)

sin(3x) = 3.sin(x) - 4.sin³(x)


= sin(3x) - cos(x) → recall the previous result

= 3.sin(x) - 4.sin³(x) - cos(x)

sin^3 (x) - cos (x)
= sin(x) - cos(x) - sin(x) cos^2(x)
= sin(x) - cos(x) (sin(x) cos(x) + 1)