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Prove that cos2α = 1-tan²α / 1+tan²α​

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Answer:

see the explanation

Explanation:

we have


cos(2\alpha)=(1-tan^2(\alpha))/(1+tan^2(\alpha))

Prove

Remember that


cos(2\alpha)=cos^2(\alpha)-sin^2(\alpha)


tan(\alpha)=(sin(\alpha))/(cos(\alpha))

substitute in the expression above


cos^2(\alpha)-sin^2(\alpha)=(1-((sin(\alpha))/(cos(\alpha)))^2)/(1+((sin(\alpha))/(cos(\alpha)))^2)


cos^2(\alpha)-sin^2(\alpha)=(((cos^2(\alpha)-sin^2(\alpha))/(cos^2(\alpha)))/(((cos^2(\alpha)+sin^2(\alpha))/(cos^2(\alpha)))}

Remember that


cos^2(\alpha)+sin^2(\alpha)=1

Simplify


cos^2(\alpha)-sin^2(\alpha)=cos^2(\alpha)-sin^2(\alpha) ---> is proved

User Coockoo
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