How do you use the sum and difference formula to simplify $\frac{1 - tan 40 tan 20}{tan 40 + tan 20}$ $(1-tan40tan20)/(tan40+tan20)$ ?

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How do you use the sum and difference formula to simplify $\frac{1 - tan 40 tan 20}{tan 40 + tan 20}$ $(1-tan40tan20)/(tan40+tan20)$ ?

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Answer 1 · 2016-09-10T10:52:30

$\frac{1 - tan 40 tan 20}{tan 40 + tan 20} = cot 60$ $(1 - tan 40 tan 20) / (tan 40 + tan 20) = cot 60$

Explanation:

The basic sum and difference formulae for $sin$ $sin$ and $cos$ $cos$ are:

$sin (α + β) = sin α cos β + sin β cos α$ $sin(alpha + beta) = sin alpha cos beta + sin beta cos alpha$

$sin (α - β) = sin α cos β - sin β cos α$ $sin(alpha - beta) = sin alpha cos beta - sin beta cos alpha$

$cos (α + β) = cos α cos β - sin α sin β$ $cos(alpha + beta) = cos alpha cos beta - sin alpha sin beta$

$cos (α - β) = cos α cos β + sin α sin β$ $cos(alpha - beta) = cos alpha cos beta + sin alpha sin beta$

There are also sum and difference formulae for $tan$ $tan$ that we can derive from the above:

$tan (α + β) = \frac{sin (α + β)}{cos (α + β)}$ $tan(alpha+beta) = sin(alpha+beta)/cos(alpha+beta)$

$tan (α + β) = \frac{sin α cos β + sin β cos α}{cos α cos β - sin α sin β}$ $color(white)(tan(alpha+beta)) = (sin alpha cos beta + sin beta cos alpha)/(cos alpha cos beta - sin alpha sin beta)$

$tan (α + β) = \frac{(sin α cos β + sin β cos α) \div (cos α cos β)}{(cos α cos β - sin α sin β) \div (cos α cos β)}$ $color(white)(tan(alpha+beta)) = ((sin alpha cos beta + sin beta cos alpha) -: (cos alpha cos beta))/((cos alpha cos beta - sin alpha sin beta) -: (cos alpha cos beta))$

$tan (α + β) = \frac{tan α + tan β}{1 - tan α tan β}$ $color(white)(tan(alpha+beta)) = (tan alpha + tan beta)/(1 - tan alpha tan beta)$

Similarly:

$tan (α - β) = \frac{tan α - tan β}{1 + tan α tan β}$ $tan(alpha-beta) = (tan alpha - tan beta)/(1 + tan alpha tan beta)$

So notice that:

$\frac{1 - tan 40 tan 20}{tan 40 + tan 20} = \frac{1}{tan (40 + 20)} = \frac{1}{tan 60} = cot 60$ $(1 - tan 40 tan 20) / (tan 40 + tan 20) = 1/(tan (40+20)) = 1/tan 60 = cot 60$

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How do you use the sum and difference formula to simplify $\frac{1 - tan 40 tan 20}{tan 40 + tan 20}$ $(1-tan40tan20)/(tan40+tan20)$ ?

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Explanation:

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How do you use the sum and difference formula to simplify 1−tan40tan20tan40+tan20(1-tan40tan20)/(tan40+tan20)?

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How do you use the sum and difference formula to simplify $\frac{1 - tan 40 tan 20}{tan 40 + tan 20}$ $(1-tan40tan20)/(tan40+tan20)$ ?