Question

Linearise cos^5(x). Help asap please
use exponential

Answer 1

Explanation:

To linearize the function cos^5(x), we can use the following identity:

cos^5(x) = (cos^2(x))^2 * cos(x) = (1 - sin^2(x))^2 * cos(x)

Now, we can use the identity e^(ix) = cos(x) + i sin(x) to rewrite the expression as follows:

(1 - sin^2(x))^2 * cos(x) = ((1 - e^(ix) * e^(-ix))/2)^2 * (e^(ix) + e^(-ix))/2

= (1/16) * (e^(4ix) - 4e^(2ix) + 6 - 4e^(-2ix) + e^(-4ix))

This expression is now linear in terms of e^(ix) and e^(-ix). Therefore, we have linearized the function cos^5(x) as:

(1/16) * (e^(4ix) - 4e^(2ix) + 6 - 4e^(-2ix) + e^(-4ix))