Question

cos 30° + sin 60° + cosec 45°÷
1+ cos 45° + sin 30° + cot 45°​

Answer 1

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\sf{\dfrac{\cos \left(30^{\circ \:}\right)+\sin \left(60^{\circ \:}\right)+coscec\left(45^{\circ \:}\right)}{1+\cos \left(45^{\circ \:}\right)+\sin \left(30^{\circ \:}\right)+\cot \left(45^{\circ \:}\right)}}$

♣ ᴀɴꜱᴡᴇʀ :

$\sf{\bigstar\:\:\:\dfrac{\cos \left(30^{\circ \:}\right)+\sin \left(60^{\circ \:}\right)+\csc \left(45^{\circ \:}\right)}{1+\cos \left(45^{\circ \:}\right)+\sin \left(30^{\circ \:}\right)+\cot \left(45^{\circ \:}\right)}}$

$\sf{Express\:with\:sin,\:cos}$

$\sf{\csc \left(45^{\circ \:}\right)=\dfrac{1}{\sin \left(45^{\circ \:}\right)}}$

$\sf{=\dfrac{\cos \left(30^{\circ \:}\right)+\sin \left(60^{\circ \:}\right)+\dfrac{1}{\sin \left(45^{\circ \:}\right)}}{1+\cos \left(45^{\circ \:}\right)+\sin \left(30^{\circ \:}\right)+\cot \left(45^{\circ \:}\right)}}}$

$\sf{\mathrm{Use\:the\:following\:identity}:\quad \cos \left(x\right)=\sin \left(90^{\circ \:}-x\right)}$

$\sf{\cos \left(30^{\circ \:}\right)=\sin \left(90^{\circ \:}-30^{\circ \:}\right)}$

$=\sf{\dfrac{\sin \left(90^{\circ \:}-30^{\circ \:}\right)+\sin \left(60^{\circ \:}\right)+\dfrac{1}{\sin \left(45^{\circ \:}\right)}}{1+\cos \left(45^{\circ \:}\right)+\sin \left(30^{\circ \:}\right)+\cot \left(45^{\circ \:}\right)}}$

$\sf{\text { Simplify } \tfrac{\sin \left(60^{\circ}\right)+\sin \left(60^{\circ}\right)+\tfrac{1}{\sin \left(45^{\circ}\right)}}{1+\cos \left(45^{\circ}\right)+\sin \left(30^{\circ}\right)+\cot \left(45^{\circ}\right)}: \tfrac{2 \sin \left(60^{\circ}\right) \sin \left(45^{\circ}\right)+1}{\sin \left(45^{\circ}\right)\left(1+\cos \left(45^{\circ}\right)+\sin \left(30^{\circ}\right)+\cot \left(45^{\circ}\right)\right)}}$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(60^{\circ \:}\right)=\dfrac{\sqrt{3}}{2}$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(45^{\circ \:}\right)=\dfrac{\sqrt{2}}{2}$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(45^{\circ \:}\right)=\dfrac{\sqrt{2}}{2}$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(45^{\circ \:}\right)=\dfrac{\sqrt{2}}{2}$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(30^{\circ \:}\right)=\dfrac{1}{2}$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \cot \left(45^{\circ \:}\right)=1$

$\sf{=\dfrac{2\cdot \frac{\sqrt{3}}{2}\cdot \dfrac{\sqrt{2}}{2}+1}{\dfrac{\sqrt{2}}{2}\left(1+\dfrac{\sqrt{2}}{2}+\dfrac{1}{2}+1\right)}}$

$\boxed{\bigstar\:\:\bf{\dfrac{2\cdot \frac{\sqrt{3}}{2}\cdot \dfrac{\sqrt{2}}{2}+1}{\dfrac{\sqrt{2}}{2}\left(1+\dfrac{\sqrt{2}}{2}+\dfrac{1}{2}+1\right)}} = \bf{\dfrac{10\sqrt{3}-2\sqrt{6}+10\sqrt{2}-4}{23}}}$