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Question
If 2 cos α = `x + 1/x` and 2 cos β = `y + 1/y`, show that `x/y + y/x = 2cos(alpha − beta)`
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Solution
Given 2 cos α = `x + 1/x` and 2 cos β = `y + 1/y`
Simplifying x² – 2x cos α + 1 = 0
Using x = `(- "b" +- sqrt("b"^2 - 4"ac"))/(2"a")`
Solving x = `(2 cos alpha +- sqrt(4 cos^2 alpha - 4))/2`
= `(2 cos alpha +- "i" 2 sin alpha)/2`
= `cosalpha +- "i" sin alpha`
if x = cos α + i sin α, then `1/x` = cos α – i sin α
Similarly y = cos β + i sin β and `1/y` = cos β – i sin β
`x/y + y/x = 2 cos (alpha - beta)`
`x/y = (cos alpha + "i "sin alpha)/(cos beta + "i "sin beta)`
= `cos(alpha - beta) + "i"sin(alpha - beta)`
`y/x = cos(alpha - beta) - "i"sin(alpha - beta)`
∴ `x/y + y/x = cos(alpha - beta) + "i" sin(alpha - beta) + cos(alpha - beta) - "i" sin (alpha - beta)`
= `2 cos (alpha - beta)`
Hence proved.
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