If $1+\sqrt{1+a}=(1+\sqrt{1-a}) \cot \alpha$ and $0

Question

If $1+\sqrt{1+a}=(1+\sqrt{1-a}) \cot \alpha$ and $0 < a < 1$, then $\sin 4 \alpha=$

Aicharya · Accepted Answer

Given $(1+\sqrt{1+a})=(1+\sqrt{1-a}) \cot \alpha$ $ \begin{aligned} & \Rightarrow \quad(1+\sqrt{1+a}) \sin \alpha=(1+\sqrt{1-a}) \cos \alpha \ & \Rightarrow \quad(\sin \alpha-\cos \alpha)=(\sqrt{1-a} \cos \alpha-\sqrt{1+a} \sin \alpha) \end{aligned} $ Squaring both side we get $ \begin{aligned} & \Rightarrow \quad 1-\sin 2 \alpha=1+a\left(\sin ^2 \alpha-\cos ^2 \alphaight)-\sqrt{1-a^2} \sin 2 \alpha \ & \Rightarrow \quad[(a \cos 2 \alpha)-\sin 2 \alpha]^2=\left[-\sqrt{1-a^2} \sin 2 \alphaight]^2 \ & \Rightarrow \quad a^2 \cos ^2 2 \alpha+\sin ^2 2 \alpha-2 a \sin 2 \alpha \cos 2 \alpha \ & =\left(1-a^2ight) \sin ^2 2 \alpha \ & \Rightarrow \quad a^2=a \sin 4 \alpha \ & \Rightarrow \quad \sin 4 \alpha=a \end{aligned} $

If $1+\sqrt{1+a}=(1+\sqrt{1-a}) \cot \alpha$ and $0 < a < 1$, then $\sin 4 \alpha=$

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