Answer
Verified
410.4k+ views
Hint – In this question use the formula that $\cos 3x = 4{\cos ^3}x - 3\cos x,\sin 3x = 3\sin x - 4{\sin ^3}x$. On simplification and application of basic algebraic identity$\left[ {{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)} \right],\left[ {{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)} \right]$, will help manipulating the equation to the final result.
Complete step-by-step answer:
Given trigonometric equation is
$\cos 3x.{\cos ^3}x + \sin 3x.{\sin ^3}x = 0$
Now as we know that $\cos 3x = 4{\cos ^3}x - 3\cos x,\sin 3x = 3\sin x - 4{\sin ^3}x$ so use this property in above equation that is
$ \Rightarrow \left( {4{{\cos }^3}x - 3\cos x} \right).{\cos ^3}x + \left( {3\sin x - 4{{\sin }^3}x} \right).{\sin ^3}x = 0$
Now simplify this equation we have,
$ \Rightarrow 4{\cos ^6}x - 3{\cos ^4}x + 3{\sin ^4}x - 4{\sin ^6}x = 0$
$ \Rightarrow 4\left( {{{\cos }^6}x - {{\sin }^6}x} \right) - 3\left( {{{\cos }^4}x - {{\sin }^4}x} \right) = 0$
$ \Rightarrow 4\left[ {{{\left( {{{\cos }^2}x} \right)}^3} - {{\left( {{{\sin }^2}x} \right)}^3}} \right] - 3\left[ {{{\left( {{{\cos }^2}x} \right)}^2} - {{\left( {{{\sin }^2}x} \right)}^2}} \right] = 0$
Now use the property that $\left[ {{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)} \right],\left[ {{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)} \right]$ so we have,
$ \Rightarrow 4\left[ {\left( {{{\cos }^2}x} \right) - \left( {{{\sin }^2}x} \right)} \right]\left[ {{{\cos }^4}x + {{\sin }^4}x + {{\cos }^2}x{{\sin }^2}x} \right] - 3\left[ {\left( {{{\cos }^2}x} \right) - \left( {{{\sin }^2}x} \right)} \right]\left[ {{{\cos }^2}x + {{\sin }^2}x} \right] = 0$
Now as we know that ${\sin ^2}x + {\cos ^2}x = 1$ so we have,
$ \Rightarrow \left( {{{\cos }^2}x - {{\sin }^2}x} \right)\left[ {4\left( {{{\cos }^4}x + {{\sin }^4}x + {{\cos }^2}x{{\sin }^2}x} \right) - 3} \right] = 0$
Now add and subtract by ${\cos ^2}x{\sin ^2}x$ in the term ${\cos ^4}x + {\sin ^4}x + {\cos ^2}x{\sin ^2}x$ we have
$ \Rightarrow \left( {\cos 2x} \right)\left[ {4\left( {{{\cos }^4}x + {{\sin }^4}x + 2{{\cos }^2}x{{\sin }^2}x - {{\cos }^2}x{{\sin }^2}x} \right) - 3} \right] = 0$, $\left[ {\because {{\cos }^2}x - {{\sin }^2}x = \cos 2x} \right]$
Now use the property ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$ we have,
$ \Rightarrow \left( {\cos 2x} \right)\left[ {4{{\left( {{{\sin }^2}x + {{\cos }^2}x} \right)}^2} - 4{{\cos }^2}x{{\sin }^2}x - 3} \right] = 0$
$ \Rightarrow \left( {\cos 2x} \right)\left[ {4 - 4{{\cos }^2}x{{\sin }^2}x - 3} \right] = 0$
$ \Rightarrow \left( {\cos 2x} \right)\left[ {1 - {{\left( {2\cos x\sin x} \right)}^2}} \right] = 0$
$ \Rightarrow \left( {\cos 2x} \right)\left[ {1 - {{\sin }^2}2x} \right] = 0$, $\left[ {\because \sin 2x = 2\sin x\cos x} \right]$
$ \Rightarrow \left( {\cos 2x} \right)\left[ {{{\cos }^2}2x} \right] = 0$, $\left[ {\because 1 - {{\sin }^2}2x = {{\cos }^2}2x} \right]$
$ \Rightarrow {\cos ^3}2x = 0$
$ \Rightarrow \cos 2x = 0 = \cos \left[ {\left( {2n + 1} \right)\dfrac{\pi }{2}} \right]$ Where, n = 0, 1, 2, 3...
Now on comparing we have,
$2x = \left( {2n + 1} \right)\dfrac{\pi }{2}$
$ \Rightarrow x = \left( {2n + 1} \right)\dfrac{\pi }{4}$, $n \in I$
So this is the required solution.
Hence option (A) is the correct answer.
Note – It is always advisable to remember basic trigonometric identity like $\sin 2x = 2\sin x\cos x$, $1 - {\sin ^2}2x = {\cos ^2}2x$ and others like $\cos 2x = {\cos ^2}x - {\sin ^2}x$, $1 + {\tan ^2}x = {\sec ^2}x$ etc. helps solving problems of this kind. The verification of $\cos 4\theta = 0 = \cos \left[ {\left( {2n + 1} \right)\dfrac{\pi }{2}} \right]$ can be provided by the fact that n in this case varies form n = 0, 1, 2... that is set of whole numbers. So if n=0, then we will have $\cos \dfrac{\pi }{2}$ which eventually will be zero. Now if we substitute 1 in place of n we get $\cos \dfrac{{3\pi }}{2}$ which is again zero. Thus $\cos \left[ {\left( {2n + 1} \right)\dfrac{\pi }{2}} \right]$ is the general value for any $\cos \theta = 0$.
Complete step-by-step answer:
Given trigonometric equation is
$\cos 3x.{\cos ^3}x + \sin 3x.{\sin ^3}x = 0$
Now as we know that $\cos 3x = 4{\cos ^3}x - 3\cos x,\sin 3x = 3\sin x - 4{\sin ^3}x$ so use this property in above equation that is
$ \Rightarrow \left( {4{{\cos }^3}x - 3\cos x} \right).{\cos ^3}x + \left( {3\sin x - 4{{\sin }^3}x} \right).{\sin ^3}x = 0$
Now simplify this equation we have,
$ \Rightarrow 4{\cos ^6}x - 3{\cos ^4}x + 3{\sin ^4}x - 4{\sin ^6}x = 0$
$ \Rightarrow 4\left( {{{\cos }^6}x - {{\sin }^6}x} \right) - 3\left( {{{\cos }^4}x - {{\sin }^4}x} \right) = 0$
$ \Rightarrow 4\left[ {{{\left( {{{\cos }^2}x} \right)}^3} - {{\left( {{{\sin }^2}x} \right)}^3}} \right] - 3\left[ {{{\left( {{{\cos }^2}x} \right)}^2} - {{\left( {{{\sin }^2}x} \right)}^2}} \right] = 0$
Now use the property that $\left[ {{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)} \right],\left[ {{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)} \right]$ so we have,
$ \Rightarrow 4\left[ {\left( {{{\cos }^2}x} \right) - \left( {{{\sin }^2}x} \right)} \right]\left[ {{{\cos }^4}x + {{\sin }^4}x + {{\cos }^2}x{{\sin }^2}x} \right] - 3\left[ {\left( {{{\cos }^2}x} \right) - \left( {{{\sin }^2}x} \right)} \right]\left[ {{{\cos }^2}x + {{\sin }^2}x} \right] = 0$
Now as we know that ${\sin ^2}x + {\cos ^2}x = 1$ so we have,
$ \Rightarrow \left( {{{\cos }^2}x - {{\sin }^2}x} \right)\left[ {4\left( {{{\cos }^4}x + {{\sin }^4}x + {{\cos }^2}x{{\sin }^2}x} \right) - 3} \right] = 0$
Now add and subtract by ${\cos ^2}x{\sin ^2}x$ in the term ${\cos ^4}x + {\sin ^4}x + {\cos ^2}x{\sin ^2}x$ we have
$ \Rightarrow \left( {\cos 2x} \right)\left[ {4\left( {{{\cos }^4}x + {{\sin }^4}x + 2{{\cos }^2}x{{\sin }^2}x - {{\cos }^2}x{{\sin }^2}x} \right) - 3} \right] = 0$, $\left[ {\because {{\cos }^2}x - {{\sin }^2}x = \cos 2x} \right]$
Now use the property ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$ we have,
$ \Rightarrow \left( {\cos 2x} \right)\left[ {4{{\left( {{{\sin }^2}x + {{\cos }^2}x} \right)}^2} - 4{{\cos }^2}x{{\sin }^2}x - 3} \right] = 0$
$ \Rightarrow \left( {\cos 2x} \right)\left[ {4 - 4{{\cos }^2}x{{\sin }^2}x - 3} \right] = 0$
$ \Rightarrow \left( {\cos 2x} \right)\left[ {1 - {{\left( {2\cos x\sin x} \right)}^2}} \right] = 0$
$ \Rightarrow \left( {\cos 2x} \right)\left[ {1 - {{\sin }^2}2x} \right] = 0$, $\left[ {\because \sin 2x = 2\sin x\cos x} \right]$
$ \Rightarrow \left( {\cos 2x} \right)\left[ {{{\cos }^2}2x} \right] = 0$, $\left[ {\because 1 - {{\sin }^2}2x = {{\cos }^2}2x} \right]$
$ \Rightarrow {\cos ^3}2x = 0$
$ \Rightarrow \cos 2x = 0 = \cos \left[ {\left( {2n + 1} \right)\dfrac{\pi }{2}} \right]$ Where, n = 0, 1, 2, 3...
Now on comparing we have,
$2x = \left( {2n + 1} \right)\dfrac{\pi }{2}$
$ \Rightarrow x = \left( {2n + 1} \right)\dfrac{\pi }{4}$, $n \in I$
So this is the required solution.
Hence option (A) is the correct answer.
Note – It is always advisable to remember basic trigonometric identity like $\sin 2x = 2\sin x\cos x$, $1 - {\sin ^2}2x = {\cos ^2}2x$ and others like $\cos 2x = {\cos ^2}x - {\sin ^2}x$, $1 + {\tan ^2}x = {\sec ^2}x$ etc. helps solving problems of this kind. The verification of $\cos 4\theta = 0 = \cos \left[ {\left( {2n + 1} \right)\dfrac{\pi }{2}} \right]$ can be provided by the fact that n in this case varies form n = 0, 1, 2... that is set of whole numbers. So if n=0, then we will have $\cos \dfrac{\pi }{2}$ which eventually will be zero. Now if we substitute 1 in place of n we get $\cos \dfrac{{3\pi }}{2}$ which is again zero. Thus $\cos \left[ {\left( {2n + 1} \right)\dfrac{\pi }{2}} \right]$ is the general value for any $\cos \theta = 0$.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The cell wall of prokaryotes are made up of a Cellulose class 9 biology CBSE
What organs are located on the left side of your body class 11 biology CBSE
Select the word that is correctly spelled a Twelveth class 10 english CBSE
a Tabulate the differences in the characteristics of class 12 chemistry CBSE