Use the​ power-reducing formulas to rewrite the expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1: 13 sin ^4 x

Answer 1

The standard power reduction identity for sine is:

sin²(x) = (1 - cos(2x))/2

First let's square both sides:

[sin²(x)]² = [(1 - cos(2x))/2]²

sin⁴(x) = (1 - 2cos(2x) + cos²(2x))/4

Then multiply both sides by 13 to get:

13sin⁴(x) = 13/4[1 - 2cos(2x) + cos²(2x)]

(A) 13sin⁴(x) = 13/4 - 13/2cos(2x) + 13/4cos²(2x)

The standard power reduction identity for cosine is:

cos²(x) = (1 + cos(2x))/2

We can multiply both sides by 13/4 to get:

13/4cos²(x) = 13/4(1 + cos(2x))/2

13/4cos²(x) = 13/8(1 + cos(2x))

13/4cos²(x) = 13/8 + 13/8cos(2x)

Then, since the expression in (A) above has 13/4cos²(2x), we can double the arguments to get:

13/4cos²(2x) = 13/8 + 13/8cos(4x)

Substituting this expression into (A) we get:

13sin⁴(x) = 13/4 - 13/2cos(2x) + 13/8 + 13/8cos(4x) then, combining 13/4 with 13/8 we get:

13sin⁴(x) = 39/8 - 13/2cos(2x) + 13/8cos(4x)