Question

Use a double angle, half angle, or power reduction formula to rewrite without exponents.

a. cos^2(5x)
b. cos^2(6x)
c. sin^4(8x)
d. None of the above

Answer 1

To rewrite the expressions without exponents, we apply the power reduction formulas, resulting in ½(1 + cos(10x)) for cos²(5x), ½(1 + cos(12x)) for cos²(6x), and for sin⁴(8x), a more complex reduction to ¼(1 - cos(16x)).

Step-by-step explanation:

We can use a power reduction formula to rewrite the expressions without exponents. For each of the trigonometric expressions given, we can apply the power reduction identities derived from the double angle formulas:

1. cos²(5x) can be rewritten as ½(1 + cos(10x)) using the power reduction formula for cosine.
2. cos²(6x) can be rewritten as ½(1 + cos(12x)) using the same principle.
3. For sin⁴(8x), we can use the identity sin²(θ) = ½(1 - cos(2θ)), and then apply the power reduction formula again to sin²(θ) to obtain ¼(1 - cos(16x))

By utilizing these identities, we transform the original expressions with exponents into forms without them.