Question

evaluate cos30/tan45 + 2sin²45/cos30 - cos30/tan30​

Answer 1

Answer:

(7-2√3)/2√3

Step-by-step explanation:

values to be substituted

Answer 2

Answer:

$\dfrac{\cos \left(30^{\circ \:}\right)}{\tan \left(45^{\circ \:}\right)}+2\cdot \dfrac{\sin ^2\left(45^{\circ \:}\right)}{\cos \left(30^{\circ \:}\right)}-\dfrac{\cos \left(30^{\circ \:}\right)}{\tan \left(30^{\circ \:}\right)}=\dfrac{7\sqrt{3}-9}{6}\quad \begin{pmatrix}\mathrm{Decimal:}&0.52073\dots \end{pmatrix}$

Step-by-step explanation:

$\dfrac{\cos \left(30^{\circ \:}\right)}{\tan \left(45^{\circ \:}\right)}+2\cdot \dfrac{\sin ^2\left(45^{\circ \:}\right)}{\cos \left(30^{\circ \:}\right)}-\dfrac{\cos \left(30^{\circ \:}\right)}{\tan \left(30^{\circ \:}\right)}$

$\gray{\mathrm{Multiply\:fractions}:\quad \:a\cdot \dfrac{b}{c}=\dfrac{a\:\cdot \:b}{c}}$

$=\dfrac{\cos \left(30^{\circ \:}\right)}{\tan \left(45^{\circ \:}\right)}+\dfrac{2\sin ^2\left(45^{\circ \:}\right)}{\cos \left(30^{\circ \:}\right)}-\dfrac{\cos \left(30^{\circ \:}\right)}{\tan \left(30^{\circ \:}\right)}$

$\gray{\mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(30^{\circ \:}\right)=\dfrac{\sqrt{3}}{2}}$

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

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

$\gray{\mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(30^{\circ \:}\right)=\dfrac{\sqrt{3}}{2}}$

$\gray{\mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(30^{\circ \:}\right)=\dfrac{\sqrt{3}}{2}}$

$\gray{\mathrm{Use\:the\:following\:trivial\:identity}:\quad \tan \left(30^{\circ \:}\right)=\dfrac{\sqrt{3}}{3}}$

$=\dfrac{\dfrac{\sqrt{3}}{2}}{1}+\dfrac{2\left(\dfrac{\sqrt{2}}{2}\right)^2}{\dfrac{\sqrt{3}}{2}}-\dfrac{\dfrac{\sqrt{3}}{2}}{\dfrac{\sqrt{3}}{3}}$

$\black{\dfrac{\dfrac{\sqrt{3}}{2}}{1}+\dfrac{2\left(\dfrac{\sqrt{2}}{2}\right)^2}{\dfrac{\sqrt{3}}{2}}-\dfrac{\dfrac{\sqrt{3}}{2}}{\dfrac{\sqrt{3}}{3}}}$

$\gray{\dfrac{\dfrac{\sqrt{3}}{2}}{1}=\dfrac{\sqrt{3}}{2}}$

$\gray{\dfrac{2\left(\dfrac{\sqrt{2}}{2}\right)^2}{\dfrac{\sqrt{3}}{2}}=\dfrac{2}{\sqrt{3}}}$

$\gray{\dfrac{\dfrac{\sqrt{3}}{2}}{\dfrac{\sqrt{3}}{3}}=\dfrac{3}{2}}$

$=\dfrac{\sqrt{3}}{2}+\dfrac{2}{\sqrt{3}}-\dfrac{3}{2}$

$\black{\mathrm{Combine\:the\:fractions\:}\dfrac{\sqrt{3}}{2}-\dfrac{3}{2}:\quad \dfrac{\sqrt{3}-3}{2}}$

$=\dfrac{\sqrt{3}-3}{2}+\dfrac{2}{\sqrt{3}}$

$\black{\mathrm{Least\:Common\:Multiplier\:of\:}2,\:\sqrt{3}:\quad 2\sqrt{3}}$

$\black{\mathrm{Adjust\:Fractions\:based\:on\:the\:LCM}}$

$\gray{\mathrm{Multiply\:each\:numerator\:by\:the\:same\:amount\:needed\:to\:multiply\:its}}$ $\gray{\mathrm{corresponding\:denominator\:to\:turn\:it\:into\:the\:LCM}\:2\sqrt{3}}$

$\gray{\mathrm{For}\:\dfrac{\sqrt{3}-3}{2}:\:\mathrm{multiply\:the\:denominator\:and\:numerator\:by\:}\:\sqrt{3}}$

$\dfrac{\sqrt{3}-3}{2}=\dfrac{\left(\sqrt{3}-3\right)\sqrt{3}}{2\sqrt{3}}$

$\gray{\mathrm{For}\:\dfrac{2}{\sqrt{3}}:\:\mathrm{multiply\:the\:denominator\:and\:numerator\:by\:}\:2}$

$\dfrac{2}{\sqrt{3}}=\dfrac{2\cdot \:2}{\sqrt{3}\cdot \:2}=\dfrac{4}{2\sqrt{3}}$

$=\dfrac{\left(\sqrt{3}-3\right)\sqrt{3}}{2\sqrt{3}}+\dfrac{4}{2\sqrt{3}}$

$\gray{\mathrm{Since\:the\:denominators\:are\:equal,\:combine\:the\:fractions}:\quad \dfrac{a}{c}\pm \dfrac{b}{c}=\dfrac{a\pm \:b}{c}}$

$=\dfrac{\left(\sqrt{3}-3\right)\sqrt{3}+4}{2\sqrt{3}}$

$\black{\mathrm{Expand}\:\left(\sqrt{3}-3\right)\sqrt{3}+4:}$

$\left(\sqrt{3}-3\right)\sqrt{3}+4$

$=\sqrt{3}\left(\sqrt{3}-3\right)+4$

$\black{\mathrm{Expand}\:\sqrt{3}\left(\sqrt{3}-3\right):\quad 3-3\sqrt{3}}$

$=3-3\sqrt{3}+4$

$\gray{\mathrm{Add\:the\:numbers:}\:3+4=7}$

$=7-3\sqrt{3}$

$\black{\mathrm{Rationalize\:}\dfrac{7-3\sqrt{3}}{2\sqrt{3}}:\quad \dfrac{7\sqrt{3}-9}{6}}$

$\dfrac{7-3\sqrt{3}}{2\sqrt{3}}$

$\gray{\mathrm{Multiply\:by\:the\:conjugate}\:\dfrac{\sqrt{3}}{\sqrt{3}}}$

$=\dfrac{\left(7-3\sqrt{3}\right)\sqrt{3}}{2\sqrt{3}\sqrt{3}}$

$\gray{\left(7-3\sqrt{3}\right)\sqrt{3}=7\sqrt{3}-9}$

$\gray{2\sqrt{3}\sqrt{3}=6}$

$=\dfrac{7\sqrt{3}-9}{6}$