Answer

Verified

471.3k+ views

Hint: - Break the angles as a sum of other angles with multiples of \[{90^ \circ }\].

Taking the L.H.S.

$ \Rightarrow \cos {24^ \circ } + \cos {55^ \circ } + \cos {125^ \circ } + \cos {204^ \circ } + \cos {300^ \circ }$ --- (1)

As we know that

$\left[ {\begin{array}{*{20}{c}}

{\cos \left( {{{180}^ \circ } - {\theta ^ \circ }} \right) = - \cos {\theta ^ \circ }} \\

{\cos \left( {{{180}^ \circ } + {\theta ^ \circ }} \right) = - \cos {\theta ^ \circ }} \\

{\cos \left( {{{360}^ \circ } - {\theta ^ \circ }} \right) = \cos {\theta ^ \circ }}

\end{array}} \right]$

So we have:

\[

\cos {125^ \circ } = \cos \left( {{{180}^ \circ } - {{55}^ \circ }} \right) = - \cos {55^ \circ } \\

\cos {204^ \circ } = \cos \left( {{{180}^ \circ } + {{24}^ \circ }} \right) = - \cos {24^ \circ } \\

\cos {300^ \circ } = \cos \left( {{{360}^ \circ } - {{60}^ \circ }} \right) = \cos {60^ \circ } \\

\]

Putting these values in equation (1) we get,

\[

\Rightarrow \cos {24^ \circ } + \cos {55^ \circ } - \cos {55^ \circ } - \cos {24^ \circ } + \cos {60^ \circ } \\

\Rightarrow \cos {60^ \circ } \\

\Rightarrow \dfrac{1}{2} = R.H.S. \\

\]

Hence the equation is proved.

Note - The following problem can also be solved by putting in the values of each of the terms, but it is easier to solve the problem by breaking the angles as a sum of other angles with multiple of \[{90^ \circ }\]. Also some of the common trigonometric identities must be remembered.

Taking the L.H.S.

$ \Rightarrow \cos {24^ \circ } + \cos {55^ \circ } + \cos {125^ \circ } + \cos {204^ \circ } + \cos {300^ \circ }$ --- (1)

As we know that

$\left[ {\begin{array}{*{20}{c}}

{\cos \left( {{{180}^ \circ } - {\theta ^ \circ }} \right) = - \cos {\theta ^ \circ }} \\

{\cos \left( {{{180}^ \circ } + {\theta ^ \circ }} \right) = - \cos {\theta ^ \circ }} \\

{\cos \left( {{{360}^ \circ } - {\theta ^ \circ }} \right) = \cos {\theta ^ \circ }}

\end{array}} \right]$

So we have:

\[

\cos {125^ \circ } = \cos \left( {{{180}^ \circ } - {{55}^ \circ }} \right) = - \cos {55^ \circ } \\

\cos {204^ \circ } = \cos \left( {{{180}^ \circ } + {{24}^ \circ }} \right) = - \cos {24^ \circ } \\

\cos {300^ \circ } = \cos \left( {{{360}^ \circ } - {{60}^ \circ }} \right) = \cos {60^ \circ } \\

\]

Putting these values in equation (1) we get,

\[

\Rightarrow \cos {24^ \circ } + \cos {55^ \circ } - \cos {55^ \circ } - \cos {24^ \circ } + \cos {60^ \circ } \\

\Rightarrow \cos {60^ \circ } \\

\Rightarrow \dfrac{1}{2} = R.H.S. \\

\]

Hence the equation is proved.

Note - The following problem can also be solved by putting in the values of each of the terms, but it is easier to solve the problem by breaking the angles as a sum of other angles with multiple of \[{90^ \circ }\]. Also some of the common trigonometric identities must be remembered.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Which are the Top 10 Largest Countries of the World?

Write a letter to the principal requesting him to grant class 10 english CBSE

10 examples of evaporation in daily life with explanations

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE