use the power - reducing formulas to rewrite the expression as an equivalent expression that does not…

Explanation:

Step1: Apply power - reducing formulas

We know that (\sin^{2}x=\frac{1 - \cos(2x)}{2}) and (\cos^{2}x=\frac{1+\cos(2x)}{2}). So, (40\sin^{2}x\cos^{2}x = 40\times\frac{1 - \cos(2x)}{2}\times\frac{1+\cos(2x)}{2}).

Step2: Simplify the expression

Using the difference of squares formula ((a - b)(a + b)=a^{2}-b^{2}), where (a = 1) and (b=\cos(2x)). We get (40\times\frac{1-\cos^{2}(2x)}{4}). Since (\sin^{2}\alpha=1-\cos^{2}\alpha), then (1-\cos^{2}(2x)=\sin^{2}(2x)). So the expression becomes (10\sin^{2}(2x)).

Step3: Apply power - reducing formula again

Using (\sin^{2}\alpha=\frac{1-\cos(2\alpha)}{2}), with (\alpha = 2x). We have (10\times\frac{1-\cos(4x)}{2}).

Step4: Final simplification

(10\times\frac{1-\cos(4x)}{2}=5(1 - \cos(4x))=5-5\cos(4x)).

Answer:

(5 - 5\cos(4x))