Question

Rewrite
`sin^25xcos^25x` simplified using power reduced formulas

Answer 1

To simplify the expression sin^2(5x)cos^2(5x) using power reduced formulas, substitute sin^2(x) = (1-cos(2x))/2 and cos^2(x) = (1+cos(2x))/2 into the expression. The simplified expression is sin^2(10x)/4.

Step-by-step explanation:

To simplify the expression sin^2(5x)cos^2(5x) using power reduced formulas, we can use the formulas:

• sin^2(x) = (1-cos(2x))/2
• cos^2(x) = (1+cos(2x))/2

Substituting these formulas into the given expression, we get:

(1-cos(10x))/2 * (1+cos(10x))/2 = (1-cos^2(10x))/4 = sin^2(10x)/4

Answer 2

Step-by-step explanation:

The power reducing formulas are given by the following:

`\sin^2 x = (1- \cos2x)/(2)`

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

We can then write the given expression as

`\sin^25x \cos^25x`

`= \left((1- \cos 2(5x))/(2) \right) \left((1+ \cos 2(5x))/(2) \right)`

`= (1)/(4)(1- \cos 10x)(1+ \cos 10x)`

`= (1)/(4)(1- \cos^2 10x)`

`= (1)/(4) \left((1- \cos 20x)/(2) \right)`

or

`\sin^25x \cos^25x= (1)/(8)(1- \cos 20x)`