1. [-1 Points] Rewrite the expression in terms of the first power of the cosine Use a graphing utility to graph both expressions to verify that both forms are the same. 8sin^4(x)cos^2(x) square

To rewrite the expression \(8\sin^4(x)\cos^2(x)\) in terms of the first power of cosine, we can use trigonometric identities.<br /><br />First, recall the Pythagorean identity:<br />\[ \sin^2(x) + \cos^2(x) = 1 \]<br /><br />We can express \(\sin^2(x)\) in terms of \(\cos^2(x)\):<br />\[ \sin^2(x) = 1 - \cos^2(x) \]<br /><br />Now, let's rewrite \(\sin^4(x)\) using \(\sin^2(x)\):<br />\[ \sin^4(x) = (\sin^2(x))^2 = (1 - \cos^2(x))^2 \]<br /><br />Next, expand \((1 - \cos^2(x))^2\):<br />\[ (1 - \cos^2(x))^2 = 1 - 2\cos^2(x) + \cos^4(x) \]<br /><br />Now substitute this back into the original expression:<br />\[ 8\sin^4(x)\cos^2(x) = 8(1 - 2\cos^2(x) + \cos^4(x))\cos^2(x) \]<br /><br />Distribute \(\cos^2(x)\) inside the parentheses:<br />\[ 8(1 - 2\cos^2(x) + \cos^4(x))\cos^2(x) = 8\cos^2(x) - 16\cos^4(x) + 8\cos^6(x) \]<br /><br />So, the rewritten expression is:<br />\[ 8\sin^4(x)\cos^2(x) = 8\cos^2(x) - 16\cos^4(x) + 8\cos^6(x) \]<br /><br />To verify that both forms are the same, you can use a graphing utility to plot both \(8\sin^4(x)\cos^2(x)\) and \(8\cos^2(x) - 16\cos^4(x) + 8\cos^6(x)\). If the graphs overlap perfectly, it confirms that the two expressions are indeed equivalent.