Answer
Verified
409.8k+ views
Hint: To solve this question, we should know a few trigonometric identities like, $\cos 2\theta =1-2{{\sin }^{2}}\theta $ and $\cos a\cos b+\sin a\sin b=\cos \left( a-b \right)$ . We should also know a few algebraic identities too like, ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$. By using these respectively, we can prove the expression given in the question.
Complete step-by-step answer:
In the given question, we have been asked to prove the expression, which is, ${{\left( \cos x-\cos y \right)}^{2}}+{{\left( \sin x-\sin y \right)}^{2}}=4{{\sin }^{2}}\left( \dfrac{x-y}{2} \right)$. For doing that, we will first consider the left hand side or the LHS, that is, ${{\left( \cos x-\cos y \right)}^{2}}+{{\left( \sin x-\sin y \right)}^{2}}$. We know that${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$, so by applying this in the LHS, we will get, $\begin{align}
& {{\cos }^{2}}x+{{\cos }^{2}}y-2\left( \cos x \right)\left( \cos y \right)+{{\sin }^{2}}x+{{\sin }^{2}}y-2\left( \sin x \right)\left( \sin y \right) \\
& \Rightarrow {{\cos }^{2}}x+{{\sin }^{2}}x+{{\cos }^{2}}y+{{\sin }^{2}}y-2\left[ \left( \cos x \right)\left( \cos y \right)+\left( \sin x \right)\left( \sin y \right) \right] \\
\end{align}$
Now, we also know that, ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$. Now, we can apply that in the above equation, so we will get the LHS as,
$\begin{align}
& 1+1-2\left[ \left( \cos x \right)\left( \cos y \right)+\left( \sin x \right)\left( \sin y \right) \right] \\
& \Rightarrow 2-2\left[ \left( \cos x \right)\left( \cos y \right)+\left( \sin x \right)\left( \sin y \right) \right] \\
\end{align}$
We also know that $\left( \cos x \right)\left( \cos y \right)+\left( \sin x \right)\left( \sin y \right)=\cos \left( x-y \right)$. By applying that in the above equation, we get the LHS as, $2-2\left[ \cos \left( x-y \right) \right]\Rightarrow 2\left[ 1-\cos \left( x-y \right) \right]$. We also know that $\cos \alpha =1-2{{\sin }^{2}}\left( \dfrac{\alpha }{2} \right)$, which implies that, $2{{\sin }^{2}}\left( \dfrac{\alpha }{2} \right)=1-\cos \alpha $ or we can also say that, $4{{\sin }^{2}}\left( \dfrac{\alpha }{2} \right)=2\left( 1-\cos \alpha \right)$. So, by substituting the same in the LHS, we get, $2\left[ 1-\cos \left( x-y \right) \right]=4{{\sin }^{2}}\left( \dfrac{x-y}{2} \right)$, which is equal to the RHS of the expression given in the question. Hence, we have proved the expression given in the question, ${{\left( \cos x-\cos y \right)}^{2}}+{{\left( \sin x-\sin y \right)}^{2}}=4{{\sin }^{2}}\left( \dfrac{x-y}{2} \right)$.
Note: While solving this type of questions, we need to remember that ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$. Sometimes, we do get stuck at a few points while proving the expressions, in that case we can also consider the RHS and simplify that side till we reach the values given in the LHS to prove the given expression.
Complete step-by-step answer:
In the given question, we have been asked to prove the expression, which is, ${{\left( \cos x-\cos y \right)}^{2}}+{{\left( \sin x-\sin y \right)}^{2}}=4{{\sin }^{2}}\left( \dfrac{x-y}{2} \right)$. For doing that, we will first consider the left hand side or the LHS, that is, ${{\left( \cos x-\cos y \right)}^{2}}+{{\left( \sin x-\sin y \right)}^{2}}$. We know that${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$, so by applying this in the LHS, we will get, $\begin{align}
& {{\cos }^{2}}x+{{\cos }^{2}}y-2\left( \cos x \right)\left( \cos y \right)+{{\sin }^{2}}x+{{\sin }^{2}}y-2\left( \sin x \right)\left( \sin y \right) \\
& \Rightarrow {{\cos }^{2}}x+{{\sin }^{2}}x+{{\cos }^{2}}y+{{\sin }^{2}}y-2\left[ \left( \cos x \right)\left( \cos y \right)+\left( \sin x \right)\left( \sin y \right) \right] \\
\end{align}$
Now, we also know that, ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$. Now, we can apply that in the above equation, so we will get the LHS as,
$\begin{align}
& 1+1-2\left[ \left( \cos x \right)\left( \cos y \right)+\left( \sin x \right)\left( \sin y \right) \right] \\
& \Rightarrow 2-2\left[ \left( \cos x \right)\left( \cos y \right)+\left( \sin x \right)\left( \sin y \right) \right] \\
\end{align}$
We also know that $\left( \cos x \right)\left( \cos y \right)+\left( \sin x \right)\left( \sin y \right)=\cos \left( x-y \right)$. By applying that in the above equation, we get the LHS as, $2-2\left[ \cos \left( x-y \right) \right]\Rightarrow 2\left[ 1-\cos \left( x-y \right) \right]$. We also know that $\cos \alpha =1-2{{\sin }^{2}}\left( \dfrac{\alpha }{2} \right)$, which implies that, $2{{\sin }^{2}}\left( \dfrac{\alpha }{2} \right)=1-\cos \alpha $ or we can also say that, $4{{\sin }^{2}}\left( \dfrac{\alpha }{2} \right)=2\left( 1-\cos \alpha \right)$. So, by substituting the same in the LHS, we get, $2\left[ 1-\cos \left( x-y \right) \right]=4{{\sin }^{2}}\left( \dfrac{x-y}{2} \right)$, which is equal to the RHS of the expression given in the question. Hence, we have proved the expression given in the question, ${{\left( \cos x-\cos y \right)}^{2}}+{{\left( \sin x-\sin y \right)}^{2}}=4{{\sin }^{2}}\left( \dfrac{x-y}{2} \right)$.
Note: While solving this type of questions, we need to remember that ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$. Sometimes, we do get stuck at a few points while proving the expressions, in that case we can also consider the RHS and simplify that side till we reach the values given in the LHS to prove the given expression.
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
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Select the correct plural noun from the given singular class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The sum of three consecutive multiples of 11 is 363 class 7 maths CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How many squares are there in a chess board A 1296 class 11 maths CBSE