Reduction Formulas

Question

Use the reduction formulas to rewrite sin^4 (8x) as an equivalent expression containing terms that do not involve powers of cosine greater than one.

Answer 1

Since the reduction formulas only talk about powers of 2, we're going to have to rewrite the equation:

sin⁴(8x) = [sin²(8x)]²

Now using the reduction formula sin²x = .5[1- cos(2x)], we have:

[sin²(8x)]² = [.5(1 - cos(2(8x)]² = [.5(1 - cos(16x)]²

Expanding this out using (a + b)² = a² + 2ab + b²:

[.5(1 - cos(16x)]² = .25(1 - 2cos(16x) + cos²(16x))

Here we have to use the reduction formula cos²x = .5[1 + cos(2x)]:

.25[1 - 2cos(16x) + cos²(16x)] = .25[1 - 2cos(16x) + .5[1 + cos(2(16x))]

Simplify:

.25[1 - 2cos(16x) + .5[1 + cos(2(16x))]

.25[1 - 2cos(16x) + .5 + .5cos(32x)]

.25[1.5 - 2cos(16x) + .5cos(32x)]