sin9θ=sinθ ⇒ sin9θ sinθ=0 ⇒ 2cos(9θ+θ2)sin(9θ θ2)=0 ⇒ cos5θsin4θ=0 ⇒ cos5θ=0, sin4θ=0 ⇒ 5θ=(2n+1)π2 or 4θ=kπ ⇒ θ=kπ4 ...

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Sum of Trigonometric Ratios in Terms of Their Product

Solve: sin9θ...

Question

Solve: $sin 9 θ = sin θ$

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Solution

$sin 9 θ = sin θ$
$\Rightarrow sin 9 θ - sin θ = 0$
$\Rightarrow 2 cos (\frac{9 θ + θ}{2}) sin (\frac{9 θ - θ}{2}) = 0$
$\Rightarrow cos 5 θ sin 4 θ = 0$
$\Rightarrow cos 5 θ = 0, sin 4 θ = 0$
$\Rightarrow 5 θ = (2 n + 1) \frac{π}{2}$ or $4 θ = k π$
$\Rightarrow θ = \frac{k π}{4}$ or $(2 n + 1) \frac{π}{10}$ where $n, k = 0, \pm 1, \pm 2, . . .$

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