`sin^(1/2)x cosx - sin^(5/2)x cosx = cos^3(x)sqrt(sin(x))` Verify the identity.

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Chapter 5, 5.2 - Problem 21 - Precalculus (3rd Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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Verify the identity.

`sin^(1/2)(x)cos(x)-sin^(5/2)(x)cos(x)=cos^3(x)sqrt(sin(x))`

Factor out the GCF `sin^(1/2)(x)cos(x).`

`sin^(1/2)(x)cos(x)[1-sin^2(x)]=cos^3(x)sqrt(sin(x))`

Use the pythagorean identity `sin^2(x)+cos^2(x)=1.`

From this identity `1-sin^2(x)=cos^2(x).`

`sin^(1/2)(x)cos(x)[cos^2(x)]=cos^3(x)sqrt(sin(x))`

`(sin(x))^(1/2)cos^3(x)=cos^3(x)sqrt(sin(x))`

`cos^3(x)sqrt(sin(x))=cos^3(x)sqrt(sin(x))`

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scisser | (Level 3) Honors

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`sin^(1/2)x cos x - sin^(5/2)x cosx = sin^(1/2)x (cos x - sin^2 x cos x) `


`= sqrt(sin x) (cos x) (1 - sin^2 x) = sqrt(sin x)(cos x)(cos^2 x) `


`= cos^3 x * sqrt(sin x) `

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