Answer

Verified

456k+ views

Hint – In this question first simplify the L.H.S by putting direct values of standard cosine angles later on simplify the R.H.S and check whether it is equal or not so use these concepts to reach the solution of the question.

Given equation is

$4\cos A\cos \left( {{{60}^0} - A} \right)\cos \left( {{{60}^0} + A} \right) = \cos 3A$

Consider L.H.S

$ \Rightarrow 4\cos A\cos \left( {{{60}^0} - A} \right)\cos \left( {{{60}^0} + A} \right)$

Now it is given that $A = {30^0}$

So, substitute the value of A in above equation, we have

\[

\Rightarrow 4\cos {30^0}\cos \left( {{{60}^0} - {{30}^0}} \right)\cos \left( {{{60}^0} + {{30}^0}} \right) \\

\Rightarrow 4\cos {30^0}\cos {30^0}\cos {90^0} \\

\]

As we know that the value of $\cos {30^0} = \dfrac{{\sqrt 3 }}{2},{\text{ }}\cos {90^0} = 0$, so substitute this value in above equation we have,

$ \Rightarrow 4\cos {30^0}\cos {30^0}\cos {90^0} = 4\left( {\dfrac{{\sqrt 3 }}{2}} \right)\left( {\dfrac{{\sqrt 3 }}{2}} \right)\left( 0 \right)$

Now as we all know multiplication with zero is always zero

Therefore

$ \Rightarrow 4\cos A\cos \left( {{{60}^0} - A} \right)\cos \left( {{{60}^0} + A} \right) = 0$……………………………… (1)

Now consider R.H.S of the given equation we have,

$ \Rightarrow \cos 3A$

Now it is given that $A = {30^0}$

So, substitute the value of A in above equation, we have

$\therefore \cos \left( {3 \times {{30}^0}} \right) = \cos {90^0}$

As we know that the value of $\cos {90^0} = 0$, so substitute this value in above equation we have,

R.H.S $ = \cos {90^0} = 0$ ……………………………………….. (2)

Now from equation (1) and (2)

L.H.S = R.H.S

So, the given statement is true.

Hence enter 1.

So, this is the required answer.

Note – In such types of questions it is advised to simplify the LHS or the RHS according to their complexity of trigonometric functions . Sometimes proving LHS = RHS needs simplification on both sides of the equation. Remember to convert dissimilar trigonometric functions to get to the final result, and check whether R.H.S is equal to L.H.S or not if yes then enter 1, which is the required answer.

__Complete step-by-step solution -__Given equation is

$4\cos A\cos \left( {{{60}^0} - A} \right)\cos \left( {{{60}^0} + A} \right) = \cos 3A$

Consider L.H.S

$ \Rightarrow 4\cos A\cos \left( {{{60}^0} - A} \right)\cos \left( {{{60}^0} + A} \right)$

Now it is given that $A = {30^0}$

So, substitute the value of A in above equation, we have

\[

\Rightarrow 4\cos {30^0}\cos \left( {{{60}^0} - {{30}^0}} \right)\cos \left( {{{60}^0} + {{30}^0}} \right) \\

\Rightarrow 4\cos {30^0}\cos {30^0}\cos {90^0} \\

\]

As we know that the value of $\cos {30^0} = \dfrac{{\sqrt 3 }}{2},{\text{ }}\cos {90^0} = 0$, so substitute this value in above equation we have,

$ \Rightarrow 4\cos {30^0}\cos {30^0}\cos {90^0} = 4\left( {\dfrac{{\sqrt 3 }}{2}} \right)\left( {\dfrac{{\sqrt 3 }}{2}} \right)\left( 0 \right)$

Now as we all know multiplication with zero is always zero

Therefore

$ \Rightarrow 4\cos A\cos \left( {{{60}^0} - A} \right)\cos \left( {{{60}^0} + A} \right) = 0$……………………………… (1)

Now consider R.H.S of the given equation we have,

$ \Rightarrow \cos 3A$

Now it is given that $A = {30^0}$

So, substitute the value of A in above equation, we have

$\therefore \cos \left( {3 \times {{30}^0}} \right) = \cos {90^0}$

As we know that the value of $\cos {90^0} = 0$, so substitute this value in above equation we have,

R.H.S $ = \cos {90^0} = 0$ ……………………………………….. (2)

Now from equation (1) and (2)

L.H.S = R.H.S

So, the given statement is true.

Hence enter 1.

So, this is the required answer.

Note – In such types of questions it is advised to simplify the LHS or the RHS according to their complexity of trigonometric functions . Sometimes proving LHS = RHS needs simplification on both sides of the equation. Remember to convert dissimilar trigonometric functions to get to the final result, and check whether R.H.S is equal to L.H.S or not if yes then enter 1, which is the required answer.

Recently Updated Pages

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

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

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

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

10 examples of law on inertia in our daily life

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

In 1946 the Interim Government was formed under a Sardar class 11 sst CBSE

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