Answer:
[tex]8cos^4}x = (3 + 4cos2x + cos4x)[/tex]
Step-by-step explanation:
Using the power reduction identity, we have that:
[tex]cos^{2}x = \frac{1}{2}(1 + cos2x)\\ \\cos^{4}x = (cos^{2}x)^2 = (\frac{1}{2}(1 + cos2x))^2\\\\cos^{4}x = \frac{1}{4} (1 + 2cos2x + cos^{2}2x)\\[/tex]
From the first line:
[tex]cos^{2}2x = \frac{1}{2}(1 + cos4x)[/tex]
Therefore:
[tex]cos^{4}x = \frac{1}{4} (1 + 2cos2x + \frac{1}{2}(1 + cos4x))\\\\cos^4}x = \frac{1}{4} (1 + 2cos2x + \frac{1}{2} + \frac{1}{2} cos4x)\\\\cos^4}x = \frac{1}{4} (\frac{3}{2} + 2cos2x + \frac{1}{2} cos4x)\\\\=> 8cos^4}x = 8 * \frac{1}{4} (\frac{3}{2} + 2cos2x + \frac{1}{2} cos4x)\\\\8cos^4}x = 2 * (\frac{3}{2} + 2cos2x + \frac{1}{2} cos4x)\\\\8cos^4}x = (3 + 4cos2x + cos4x)[/tex]
