`sec^6(x)(sec(x)tan(x)) - sec^4(x)(sec(x)tan(x)) = sec^5(x)tan^3(x)` Verify the identity.

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Chapter 5, 5.2 - Problem 22 - Precalculus (3rd Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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Verify the identity.

`sec^6(x)(sec(x)tan(x))-sec^4(x)(sec(x)tan(x))=sec^5(x)tan^3(x)`

Factor out the GCF `sec^4(x)(sec(x)tan(x))`

`sec^4(x)(sec(x)tan(x))[sec^2(x)-1]=sec^5(x)tan^3(x)`

Use the pythagorean identity `1+tan^2(x)=sec^2(x).`

From this identity `sec^2(x)-1=tan^2(x).`

`sec^4(x)(sec(x)tan(x))[tan^2(x)]=sec^5(x)tan^3(x)`

`sec^4(x)sec(x)tan(x)tan^2(x)=sec^5(x)tan^3(x)`

`sec^5(x)tan^3(x)=sec^5(x)tan^3(x)`

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