Question

Use a power reduction identity to simplify 8cos4 x .

Answer 1

Explanation:

Answer 2

`8cos^4}x = (3 + 4cos2x + cos4x)`

Explanation:

Using the power reduction identity, we have that:

`cos^(2)x = (1)/(2)(1 + cos2x)\\ \\cos^(4)x = (cos^(2)x)^2 = ((1)/(2)(1 + cos2x))^2\\\\cos^(4)x = (1)/(4) (1 + 2cos2x + cos^(2)2x)\\`

From the first line:

`cos^(2)2x = (1)/(2)(1 + cos4x)`

Therefore:

`cos^(4)x = (1)/(4) (1 + 2cos2x + (1)/(2)(1 + cos4x))\\\\cos^4}x = (1)/(4) (1 + 2cos2x + (1)/(2) + (1)/(2) cos4x)\\\\cos^4}x = (1)/(4) ((3)/(2) + 2cos2x + (1)/(2) cos4x)\\\\=> 8cos^4}x = 8 * (1)/(4) ((3)/(2) + 2cos2x + (1)/(2) cos4x)\\\\8cos^4}x = 2 * ((3)/(2) + 2cos2x + (1)/(2) cos4x)\\\\8cos^4}x = (3 + 4cos2x + cos4x)`