We know thar, A 2 = 1- 2 put A=45^ 221^ 2 = 1- 45^ 2 since 221 2 ,is positive = 1-1 2 2 221^ 2 = 2-1 22 and A 2 1+ 2 put A 45^ 221^ 2 = 1+ 45^ 2 = 1-1 2 2 221^ 2 = 2+1 22 Now, 221^ 2 = 221^ 2 221^ 2 = 2+1 22 22 2-1 = 2+1 2-1 Rationalizing denominator, = 2+1 2+1 2-1 2+1 = (2+1)^2 2-1 221^ 2 =2+1

Prove that: 8 =2+1

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Question

Prove that:
$\cot\frac{\pi}{8}=\sqrt{2}+1$

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Answer

We know thar,
$\sin\frac{\text{A}}{2}=\pm\sqrt{\frac{1-\cos\text{A}}{2}}$
$\text{put}\ \text{A}=45^\circ$
$\sin22\frac{1^\circ}{2}=\sqrt{\frac{1-\cos45^\circ}{2}}$ $\Big\{\text{since}\sin22\frac{1}{2},\text{is positive}\Big\}$
$=\sqrt{\frac{1-\frac{1}{2}}{2}}$
$\sin22\frac{1^\circ}{2}=\sqrt{\frac{\sqrt{2}-1}{2\sqrt{2}}}$
and
$\cos\frac{\text{A}}{2}\pm\sqrt{\frac{1+\cos\text{A}}{2}}$
$\text{put}\ \text{A}\ 45^\circ$
$\cos22\frac{1^\circ}{2}=\sqrt{\frac{1+\cos45^\circ}{2}}$
$=\sqrt{\frac{1-\frac{1}{2}}{2}}$
$\cos22\frac{1^\circ}{2}=\sqrt{\frac{\sqrt{2}+1}{2\sqrt{2}}}$
Now,
$\cot22\frac{1^\circ}{2}=\frac{\cos22\frac{1^\circ}{2}}{\sin22\frac{1^\circ}{2}}$
$=\sqrt{\frac{\sqrt{2}+1}{2\sqrt{2}}\times\frac{2\sqrt{2}}{\sqrt{2-1}}}$
$=\sqrt{\frac{\sqrt{2}+1}{\sqrt{2}-1}}$
Rationalizing denominator,
$=\sqrt{\frac{\sqrt{2}+1\times\sqrt{2}+1}{\sqrt{2}-1\times\sqrt{2}+1}}$
$=\sqrt{\frac{(\sqrt{2}+1)^2}{2-1}}$
$\cot22\frac{1^\circ}{2}=\sqrt{2}+1$

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