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Question

Solve :cos24o+cos55o+cos125o+cos204o+cos300o=12

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Solution

cos24o+cos55o+cos125o+cos204o+cos300o....(1)
write the above equation as
$=\cos { { 24 }^{ o } } +\sin { \left( { 90 }^{ 0 }-{ 55 }^{ 0 } \right) } +\cos { \left( { 90 }^{ 0 }-{ 35 }^{ 0 } \right) } +\cos { \left( { 180 }^{ 0 }-{ 24 }^{ 0 } \right) } +\cos { \left( { 360 }^{ 0 }-{ 60 }^{ 0 } \right) } ....(2)$
We know that $\cos { \theta } =\sin { \left( 90-\theta \right) } $
From the properties of trignometry
$\cos { \left( 180+\theta \right) } =\cos { \theta } ...(3)$
$\cos { \left( 90+\theta \right) } =\sin { \theta } ....(4)$
$\cos { \left( 360-\theta \right) } =\cos { \theta } ...(5)$
by using 3,4,5 and substitute then in (2) we get
cos24o+sin35o+(sin35o)+(cos24o)+cos60o
=cos24ocos24o+sin35osin35o+cos60o=12

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