Question

Prove that cos5x = 16cos^5x - 20cos^3x + ___.

A) 10cosx
B) 10cos^2x
C) 10cos4x
D) 10cos^4x

Answer 1

To prove the equation, use the formula cos(a + b) = cos(a)cos(b) - sin(a)sin(b). Substitute the values and simplify to find the missing term.

Step-by-step explanation:

To prove the equation cos(5x) = 16cos^5(x) - 20cos^3(x) + ___, we can use the formula cos(a + b) = cos(a)cos(b) - sin(a)sin(b).

Let's apply this formula to the equation:

cos(5x) = cos(3x + 2x) = cos(3x)cos(2x) - sin(3x)sin(2x).

We can substitute cos(3x) = (4cos^3(x) - 3cos(x)) and sin(3x) = 3sin(x) - 4sin^3(x) into the equation to get:

cos(5x) = (4cos^3(x) - 3cos(x))(2cos^2(x) - 1) - (3sin(x) - 4sin^3(x))(2sin(x)cos(x)).

Simplifying further, we get cos(5x) = 16cos^5(x) - 20cos^3(x) + _____.