Prove that cos^2A+cos^2(A+120)+cos^2(A-120)=3/2
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Posted by Himanshu Goyal 6 years, 10 months ago
- 2 answers
Rashmi Bajpayee 6 years, 10 months ago
{tex}{\cos ^2}2{\rm{A}} + {\cos ^2}2\left( {{\rm{A}} + {{120}^ \circ }} \right) + {\cos ^2}2\left( {{\rm{A}} - {{120}^ \circ }} \right) = {3 \over 2}{/tex}
We know that
{tex}{\cos ^2}{\rm{A}} = {{1 + \cos 2{\rm{A}}} \over 2}{/tex} ..............(i)
{tex}{\cos ^2}2\left( {{\rm{A}} + {{120}^ \circ }} \right) = {{1 + \cos \left( {{\rm{2A}} + {{240}^ \circ }} \right)} \over 2}{/tex} ..............(ii)
And {tex}{\cos ^2}2\left( {{\rm{A}} - {{120}^ \circ }} \right) = {{1 + \cos \left( {{\rm{2A}} - {{240}^ \circ }} \right)} \over 2}{/tex}..............(iii)
Adding all these three equations,
{tex}{\cos ^2}2{\rm{A}} + {\cos ^2}2\left( {{\rm{A}} + {{120}^ \circ }} \right) + {\cos ^2}2\left( {{\rm{A}} - {{120}^ \circ }} \right){/tex} = {tex}{3 \over 2} + {{1 + \cos 2{\rm{A}}} \over 2} + {{1 + \cos \left( {{\rm{2A + }}{{240}^ \circ }} \right)} \over 2} + {{1 + \cos \left( {{\rm{2A}} - {{240}^ \circ }} \right)} \over 2}{/tex}
= {tex}{{3 + 1 + \cos 2{\rm{A + }}1 + \cos \left( {{\rm{2A + }}{{240}^ \circ }} \right) + 1 + \cos \left( {{\rm{2A}} - {{240}^ \circ }} \right)} \over 2}{/tex}
= {tex}{{3 + 3 + \cos 2{\rm{A}} + \cos \left( {{\rm{2A + }}{{240}^ \circ }} \right) + \cos \left( {{\rm{2A}} - {{240}^ \circ }} \right)} \over 2}{/tex}
= {tex}{{3 + 3 + \cos 2{\rm{A}} + 2\cos 2{\rm{A}}.\cos {{240}^ \circ }} \over 2}{/tex}
= {tex}{{3 + 3 + \cos 2{\rm{A}} - \cos 2{\rm{A}}} \over 2}{/tex}
= {tex}{6 \over 2}{/tex}
{tex}\Rightarrow{/tex} {tex}{\cos ^2}2{\rm{A}} + {\cos ^2}2\left( {{\rm{A}} + {{120}^ \circ }} \right) + {\cos ^2}2\left( {{\rm{A}} - {{120}^ \circ }} \right) = {3 \over 2}{/tex}
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James Black 5 years, 10 months ago
0Thank You