If tan -n tan - show that -n-1 sin 2 -n-1 sin 2 -

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Arithmetic Mean

If tan α+θ=n ...

Question

If tan(α+θ) =n tan(α-θ) show that : (n+1)sin 2θ =(n-1)sin2α

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If $t a n x + t a n (x + \frac{π}{3}) + t a n (x + \frac{2 π}{3}) = 3,$ prove that $\frac{3 t a n x - t a n^{3} x}{1 - 3 t a n^{2} x = 1.}$

If $s i n θ = n s i n (θ + 2 α),$ prove that $t a n (θ + α) = \frac{1 + n}{1 - n} t a n α .$

Which of the following statements are correct?

1 . if sin θ = sin α ⇒ θ = nπ + $(- 1)^{n}$ α where α ∈ [ - $\frac{π}{2}$ , $\frac{π}{2}$ ] n ∈ I

2 . if cos θ = cos α ⇒ 2nπ±α where α ∈ [0,π] n ∈ I

3 . if tan θ = tan α ⇒ θ = nπ + α where α ∈ (- $\frac{π}{2}$ , $\frac{π}{2}$ ) n ∈ I

m s i n θ = n c o s θ

\frac{t a n θ + c o t θ}{t a n θ - c o t θ} = \frac{n s i n θ + m c o s θ}{n s i n θ - m c o s θ} = \frac{n^{2} + m^{2}}{n^{2} - m^{2}}

tan θ = tan α, \forall α \in (- \frac{π}{2}, \frac{π}{2})

θ = 2 n π + α, n \in Z .

α < \frac{π}{4 (n - 1)}

tan n

α > n tan α

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