By expressing 3x as (2x + x) prove that
cos3x = 4cos^3x - 3cosx
cos(3x)
= cos(2x + x)
= cos2xcosx - sin2xsinx
= (2cos^2 x -1)cosx - (2sinxcosx)sinx
= 2cos^3 x - cosx - 2sin^2 x (cosx)
= 2 cos^3x - cosx - 2(1 - cos^2 x)(cosx)
= 2 cos^3x - cosx - 2cosx + 2cos^3x
= 4cos^3x - 3cosx
= RS
Nice!
To prove that cos(3x) = 4cos^3(x) - 3cos(x) using the expression 3x = 2x + x, we'll use the trigonometric identity known as the triple angle formula:
cos(3x) = cos(2x + x).
Using the angle sum formula for cosine, we can expand cos(2x + x) as:
cos(2x + x) = cos(2x)cos(x) - sin(2x)sin(x).
Now, we need to express cos(2x) and sin(2x) in terms of cos(x) and sin(x). We can use the double angle formulas for cosine and sine:
cos(2x) = cos^2(x) - sin^2(x),
sin(2x) = 2sin(x)cos(x).
Replacing cos(2x) and sin(2x) in the expanded expression, we have:
cos(2x)cos(x) - sin(2x)sin(x) = (cos^2(x) - sin^2(x))cos(x) - 2sin(x)cos(x)sin(x).
Now, we simplify and rearrange the terms:
cos(2x)cos(x) - sin(2x)sin(x) = cos^3(x) - sin^2(x)cos(x) - 2sin(x)cos^2(x).
Since sin^2(x) + cos^2(x) = 1, we can rewrite the expression as:
cos^3(x) - (1 - cos^2(x))cos(x) - 2sin(x)cos^2(x).
Expanding the expression, we have:
cos^3(x) - cos(x) + cos^3(x) - 2sin(x)cos^2(x).
Combining like terms, we get:
2cos^3(x) - cos(x) - 2sin(x)cos^2(x).
Now, we can express sin(x) in terms of cos(x) using the trigonometric identity sin^2(x) = 1 - cos^2(x):
2cos^3(x) - cos(x) - 2(1 - cos^2(x))cos^2(x).
Simplifying further, we have:
2cos^3(x) - cos(x) - 2cos^2(x) + 2cos^4(x).
Finally, factoring out cos(x) from each term, we get:
cos(x)(2cos^2(x) - 1 - 2cos(x) + 2cos^3(x)).
Using the identity 2cos^2(x) - 1 = cos(2x), we can further simplify:
cos(x)(cos(2x) - 2cos(x) + 2cos^3(x)).
Now, we observe that the expression inside the parentheses is equal to cos(3x), so we can rewrite it as:
cos(x)cos(3x).
Since cos(x)cos(3x) is equal to cos(3x)cos(x), we conclude:
cos(3x) = 4cos^3(x) - 3cos(x).
Therefore, we have proved that cos(3x) = 4cos^3(x) - 3cos(x) using the expression 3x = 2x + x.