Question

Rewrite the expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1.

Answer 1

The expression becomes;

`(3)/(8)-(1)/(2)\cos 2x+(1)/(8)\cos 4x`

Step-by-step explanation:

Given the trigonometric expression;

`\sin ^4x`

Simplifying and rewriting the expression;

Recall that;

`\begin{gathered} \cos 2x=1-2\sin ^2x \\ \sin ^2x=(1-\cos 2x)/(2) \end{gathered}`

So, the expression becomes;

`\begin{gathered} \sin ^4x=(\sin ^2x)(\sin ^2x) \\ =((1-\cos2x)/(2))((1-\cos2x)/(2)) \\ =((1-2\cos2x+\cos^22x)/(4)) \\ =(1)/(4)-(2)/(4)\cos 2x+(1)/(4)\cos ^22x \end{gathered}`

Also;

`\begin{gathered} \cos 4x=2\cos ^22x-1 \\ \cos ^22x=(\cos 4x+1)/(2) \end{gathered}`

substituting to the above expression;

`\begin{gathered} =(1)/(4)-(2)/(4)\cos 2x+(1)/(4)((\cos4x+1)/(2)) \\ =(1)/(4)-(1)/(2)\cos 2x+(1)/(8)\cos 4x+(1)/(8) \\ =(1)/(4)+(1)/(8)-(1)/(2)\cos 2x+(1)/(8)\cos 4x \\ =(3)/(8)-(1)/(2)\cos 2x+(1)/(8)\cos 4x \end{gathered}`

Therefore, the expression becomes;

`(3)/(8)-(1)/(2)\cos 2x+(1)/(8)\cos 4x`