Trigonometry! #104

Geometry Level 3

If sin α + csc α = 2 \sin\alpha+\csc\alpha=2 then sin n α + csc n α = ? \sin^{n}\alpha+\csc^{n}\alpha=?

This problem is part of the set Trigonometry .

n n n ( sin α + csc α ) n(\sin\alpha+\csc\alpha) n sin α + csc α \frac{n}{\sin\alpha+\csc\alpha} 2 2

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1 solution

Pranjal Jain
Feb 10, 2015

Let x = sin α x=\sin\alpha

x + 1 x = 2 x 2 2 x + 1 = 0 x = 1 x+\dfrac{1}{x}=2\\\Rightarrow x^2-2x+1=0\\\Rightarrow x=1

sin n α + csc n α = 1 n + 1 n = 1 + 1 = 2 \sin^n\alpha+\csc^n\alpha=1^n+1^n=1+1=\boxed{2}

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