Questions: y=4 cos^2 x (a) Rewrite the function as a single trigonometric function raised to the first power.

Solution Steps

To rewrite the function \( y = 4 \cos^2 x \) as a single trigonometric function raised to the first power, we can use the Pythagorean identity for cosine and sine. The identity states that \( \cos^2 x = 1 - \sin^2 x \). By substituting this identity into the given function, we can express \( y \) in terms of \( \sin x \).

1. Use the Pythagorean identity \( \cos^2 x = 1 - \sin^2 x \).
2. Substitute \( \cos^2 x \) in the given function \( y = 4 \cos^2 x \) with \( 1 - \sin^2 x \).
3. Simplify the expression to get \( y \) in terms of \( \sin x \).

We start with the function given by

\[ y = 4 \cos^2 x. \]

Using the Pythagorean identity

\[ \cos^2 x = 1 - \sin^2 x, \]

we can substitute this into the function:

\[ y = 4 (1 - \sin^2 x). \]

Now, we simplify the expression:

\[ y = 4 - 4 \sin^2 x. \]

This shows that the function can be expressed in terms of \( \sin x \).

Final Answer