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Wednesday, April 29, 2015

Prove tan55tan65tan75=tan85 (Method I)

Let's begin with the left side:

tan(55)tan(65)tan(75)

Using the product to sum identity for the tangent function:

tan(α)tan(β)=cos(αβ)cos(α+β)cos(αβ)cos(α+β)

we may write:

tan(65)tan(55)=cos(10)cos(120)cos(10)+cos(120)

Given that cos(120)=12 we now have:

tan(65)tan(55)=cos(10)+12cos(10)12=2cos(10)+12cos(10)1

Hence, we may write:

tan(55)tan(65)tan(75)=2cos(10)+12cos(10)1tan(75)

Using the identity:

tan(α)=sin(α)cos(α)

we have:

tan(55)tan(65)tan(75)=(2cos(10)+1)sin(75)(2cos(10)1)cos(75)

Distributing, there results:

tan(55)tan(65)tan(75)=2sin(75)cos(10)+sin(75)2cos(75)cos(10)cos(75)

Using the product-to-sum identity:

2sin(α)cos(β)=sin(α+β)+sin(αβ)

we have:

2sin(75)cos(10)=sin(85)+sin(65)

Using the product-to-sum identity:

2cos(α)cos(β)=cos(α+β)+cos(αβ)

we have:

2cos(75)cos(10)=cos(85)+cos(65)

And now we may state:

tan(55)tan(65)tan(75)=sin(85)+sin(65)+sin(75)cos(85)+cos(65)cos(75)

Rearranging, we have:

tan(55)tan(65)tan(75)=sin(85)+sin(75)+sin(65)cos(85)(cos(75)cos(65))

Using the sum-to-product identity:

sin(α)+sin(β)=2sin(α+β2)cos(αβ2)

we have:

sin(75)+sin(65)=2sin(70)cos(5)

Using the sum-to-product identity:

cos(α)cos(β)=2sin(α+β2)sin(αβ2)

we have:

(cos(75)cos(65))=2sin(70)sin(5)

Thus, we now have:

tan(55)tan(65)tan(75)=sin(85)+2sin(70)cos(5)cos(85)+2sin(70)sin(5)

Using the co-function identities:

cos(α)=sin(90α)

sin(α)=cos(90α)

we have:

cos(5)=sin(85)

sin(5)=cos(85)

and we now may write:

tan(55)tan(65)tan(75)=sin(85)+2sin(70)sin(85)cos(85)+2sin(70)cos(85)

Factoring, we get:

tan(55)tan(65)tan(75)=sin(85)(1+2sin(70))cos(85)(1+2sin(70))

Dividing out common factors, we now have:

tan(55)tan(65)tan(75)=sin(85)cos(85)

Using the identity:

tan(α)=sin(α)cos(α)

we have:

tan(55)tan(65)tan(75)=tan(85)

Shown as desired.

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