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Product of Trigonometric Ratios in Terms of Their Sum

if 2 cosθ ...

Question

if $2 c o s θ = x = \frac{1}{x} a n d 2 c o s ϕ = y + \frac{1}{y}$ etc., then prove that :
$x y z . . . . + \frac{1}{x y z . . . .} = 2 c o s (θ + ϕ + . . . .)$

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Solution

From the given relations
$x = c o s θ + i s i n θ = e^{i θ}$
$y = c o s ϕ + i s i n ϕ = e^{i ϕ}$
$x y z . . . . = e^{i (θ + ϕ + ψ + . . . .)}$
$\frac{1}{x y z . . . .} = e^{- i (θ + ϕ + ψ + . . . .)}$
$∴ x y z . . . . + \frac{1}{x y z . . . .} 2 c o s (θ + ϕ + ψ + . . . .)$

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Product of Trigonometric Ratios in Terms of Their Sum

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