Question

answer ..,..............​

Answer 1

$\color{red}\huge{\underline{\underline{\bf GIVEN\dag}}}$

• $2 \cos( \alpha - \beta ) = \sqrt{3}$
• $\sin( \alpha + \beta ) = \frac{ \sqrt{3} }{2}$

$\color{red}\huge{\underline{\underline{\bf SOLUTION\dag}}}$

$2 \cos( \alpha - \beta ) = \sqrt{3}$

$\implies \cos( \alpha - \beta ) = \frac{ \sqrt{3} }{2}$

$\bf but$

$\blue{ \frac{ \sqrt{3} }{2} = \cos(30)}$

$\implies \cos( \alpha - \beta ) = \cos(30)$

$\therefore \alpha - \beta = 30 \: ......(1)$

$\bf similarly$

$\sin(\alpha + \beta ) =\frac{\sqrt{3}}{2}$

$\bf but$

$\blue{\sin(60)=\frac{\sqrt{3}}{2}}$

$\implies \sin(\alpha + \beta ) =\sin(60)$

$\therefore \alpha + \beta = 60 \: ......(2)$

$\bf solving\:(1)\:and\:(2)\:we\:get$

$\alpha=45$

$\beta=15$

$\bf Now$

$\cot(\alpha +2 \beta)=\cot(45+2(15))$

$\cot(\alpha +2 \beta)=\cot(45+30)$

$\cot(\alpha +2 \beta)=\cot(75)$

$\orange{\boxed{\therefore\cot(\alpha +2 \beta)=2-\sqrt{3}}}$

HOPE THIS HELPS!!