If 2 cos α = `x + 1/x` and 2 cos β = `y + 1/y`, show that `x^"m" y^"n" + 1/(x^"m" y^"n")` = 2 cos(mα – nβ)

= (cos mα + sin mα)(cos nβ + i sin nβ) = cos(mα + nβ) + i sin(mα + nβ) `1/(x^"m" y^"n")` = cos(mα + nβ) – i sin(mα + nβ) &there4; `x^"m" y^"n" + 1/(x^"m" y^"n")` = 2 cos(mα – nβ) Hence proved

If 2 cos α = x+1x and 2 cos β = y+1y, show that mnmnxmyn+1xmyn = 2 cos(mα – nβ) - Mathematics

Question

If 2 cos α = $x + \frac{1}{x}$ and 2 cos β = $y + \frac{1}{y}$ , show that $x^{m} y^{n} + \frac{1}{x^{m} y^{n}}$ = 2 cos(mα – nβ)

Sum

Solution

= (cos mα + sin mα)(cos nβ + i sin nβ)

= cos(mα + nβ) + i sin(mα + nβ)

$\frac{1}{x^{m} y^{n}}$ = cos(mα + nβ) – i sin(mα + nβ)

∴ $x^{m} y^{n} + \frac{1}{x^{m} y^{n}}$ = 2 cos(mα – nβ)

Hence proved

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de Moivre’s Theorem and Its Applications

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Q 4. (iv)Q 4. (iii)Q 5

Chapter 2: Complex Numbers - Exercise 2.8 [Page 92]

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Samacheer Kalvi Mathematics - Volume 1 and 2 [English] Class 12 TN Board

Chapter 2 Complex Numbers
Exercise 2.8 | Q 4. (iv) | Page 92

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If 2 cos α = x+1x and 2 cos β = y+1y, show that mnmnxmyn+1xmyn = 2 cos(mα – nβ) - Mathematics

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