Question

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Answer 1

Answer:

$1)cos\alpha -sin\alpha =\sqrt{2} \ sin\alpha \\\\cot\alpha -1=\sqrt{2} \\\\cot\alpha =\sqrt{2} +1\\\\\alpha =cot^{-1}( \sqrt{2} +1)\\\\\frac{\alpha }{2} =\frac{cot^{-1}( \sqrt{2} +1)}{2}$

$2) cos\alpha -sin\alpha =\sqrt{2}sin\alpha \\\\sin\alpha (\sqrt{2}+1)=cos\alpha \\\\ sin\alpha =\frac{cos\alpha }{\sqrt{2}+1} \\\\cos\alpha +sin\alpha =\sqrt{2} \ cos\alpha \\\\LHS\\\\cos\alpha +sin\alpha \\\\=cos\alpha + \frac{cos\alpha}{\sqrt{2}+1 } \\\\= cos\alpha+\frac{cos\alpha (\sqrt{2}-1) }{(\sqrt{2}+1)(\sqrt{2}-1)} \\\\=cos\alpha +\sqrt{2}cos\alpha -cos\alpha \\\\=\sqrt{2}cos\alpha \\\\ therefore\ LHS=RHS$