Question

Use the formulas for lowering powers to rewrite the expression

in terms of the first power of cosine, as in Example 4.
cos6(x)

Answer 1

The expression cos6(x) can be rewritten as 1/2 + 1/2 * cos(6x). To rewrite cos6(x) in terms of the first power of cosine, we can use the formula cos²(x) = (1 + cos(2x))/2 repeatedly.

Step-by-step explanation:

To express cos⁶(x) in terms of the first power of cosine, we utilize the trigonometric identity cos²(x) = (1 + cos(2x))/2 iteratively.

The step-by-step process unfolds as follows:

\cos^6(x) &= \cos^2(3x)\\

&= \frac{1 + \cos(6x)}{2}

\end{align*} \]

Applying the identity once more, we get:

\[ \cos(6x) = \cos^2(3x) = \frac{1 + \cos(6x)}{2} \]

This leads to the simplified expression:

\[ \cos^6(x) = \frac{1}{2} + \frac{1}{2} \cos(6x) \]

So, the expression cos⁶(x) can be rephrased as 1/2 + 1/2 * cos(6x), demonstrating its relationship with the first power of cosine through the systematic application of trigonometric identities.

Therefore, the expression cos6(x) can be rewritten as 1/2 + 1/2 * cos(6x).

Answer 2

To rewrite cos6(x) in terms of the first power of cosine, we can use the formula for lowering powers. Using the formula, cos6(x) can be rewritten as (1/2)(cos5(x) + cos7(x)).

Step-by-step explanation:

To rewrite the expression cos6(x) in terms of the first power of cosine, we can use the formula for lowering powers. The formula states that cos^n(x) = (1/2)(cos((n-1)x) + cos((n+1)x)).

Using this formula, we can rewrite cos6(x) as (1/2)(cos5(x) + cos7(x)).

So, cos6(x) = (1/2)(cos5(x) + cos7(x)).