
If $A+B+C=\pi $ , prove that $\sin 2A-\sin 2B+\sin 2C=4\cos A\sin B\cos C$.
Answer
587.1k+ views
Hint: For solving this question we will use some trigonometric formula like formula for $\sin C-\sin D$ and $\sin \left( \pi -\theta \right)$ for simplifying the term written on the left-hand side. After that, we will prove it equal to the term on the right-hand side.
Complete step-by-step answer:
Given:
It is given that if $A+B+C=\pi $ and we have to prove the following equation:
$\sin 2A-\sin 2B+\sin 2C=4\cos A\sin B\cos C$
Now, before we proceed we should know the following four formulas:
$\begin{align}
& \sin C-\sin D=2\cos \left( \dfrac{C+D}{2} \right)\sin \left( \dfrac{C-D}{2} \right)..................\left( 1 \right) \\
& A+B+C=\pi \\
& \Rightarrow A+B=\pi -C \\
& \Rightarrow \sin \left( A+B \right)=\sin \left( \pi -C \right)=\sin C..................................\left( 2 \right) \\
& \Rightarrow \cos \left( A+B \right)=\cos \left( \pi -C \right)=-\cos C..............................\left( 3 \right) \\
& \sin 2\theta =2\sin \theta \cos \theta ..................................................\left( 4 \right) \\
\end{align}$
Now, we will be using the above four formulas to simplify the term on the left-hand side to prove the desired result.
Now, L.H.S is equal to $\sin 2A-\sin 2B+\sin 2C$ so, using the formula from equation (1).
Then,
$\begin{align}
& \sin 2A-\sin 2B+\sin 2C \\
& \Rightarrow 2\cos \left( \dfrac{2A+2B}{2} \right)\sin \left( \dfrac{2A-2B}{2} \right)+\sin 2C
\\
& \Rightarrow 2\cos \left( A+B \right)\sin \left( A-B \right)+\sin 2C \\
\end{align}$
Now, using the formula from equation (3) and equation (4) in the above equation. Then,
$\begin{align}
& 2\cos \left( A+B \right)\sin \left( A-B \right)+\sin 2C \\
& \Rightarrow -2\cos C\sin \left( A-B \right)+2\sin C\cos C \\
& \Rightarrow 2\cos C\left( \sin C-\sin \left( A-B \right) \right) \\
\end{align}$
Now, using the formula from equation (2) and equation (1) in the above equation. Then,
$\begin{align}
& 2\cos C\left( \sin C-\sin \left( A-B \right) \right) \\
& \Rightarrow 2\cos C\left( \sin \left( A+B \right)-\sin \left( A-B \right) \right) \\
& \Rightarrow 2\cos C\left( 2\cos \left( \dfrac{A+B+A-B}{2} \right)\sin \left(
\dfrac{A+B-A+B}{2} \right) \right) \\
& \Rightarrow 2\cos C\left( 2\cos A\sin B \right) \\
& \Rightarrow 4\cos A\sin B\cos C \\
\end{align}$
Now, from the above result, we can say that $\sin 2A-\sin 2B+\sin 2C=4\cos A\sin B\cos C$
Thus, $L.H.S=R.H.S$ .
Hence, proved.
Note: Here, the student should first understand what we have to prove in the question and then proceed in a stepwise manner while solving. We should also try to make use of trigonometric results like $\sin \left( \pi -\theta \right)=\sin \theta $ for making equations that will help us further in the solution. Moreover, the formula $\sin C-\sin D$ should be applied correctly with proper values and avoid making calculation mistakes while solving the problem.
Complete step-by-step answer:
Given:
It is given that if $A+B+C=\pi $ and we have to prove the following equation:
$\sin 2A-\sin 2B+\sin 2C=4\cos A\sin B\cos C$
Now, before we proceed we should know the following four formulas:
$\begin{align}
& \sin C-\sin D=2\cos \left( \dfrac{C+D}{2} \right)\sin \left( \dfrac{C-D}{2} \right)..................\left( 1 \right) \\
& A+B+C=\pi \\
& \Rightarrow A+B=\pi -C \\
& \Rightarrow \sin \left( A+B \right)=\sin \left( \pi -C \right)=\sin C..................................\left( 2 \right) \\
& \Rightarrow \cos \left( A+B \right)=\cos \left( \pi -C \right)=-\cos C..............................\left( 3 \right) \\
& \sin 2\theta =2\sin \theta \cos \theta ..................................................\left( 4 \right) \\
\end{align}$
Now, we will be using the above four formulas to simplify the term on the left-hand side to prove the desired result.
Now, L.H.S is equal to $\sin 2A-\sin 2B+\sin 2C$ so, using the formula from equation (1).
Then,
$\begin{align}
& \sin 2A-\sin 2B+\sin 2C \\
& \Rightarrow 2\cos \left( \dfrac{2A+2B}{2} \right)\sin \left( \dfrac{2A-2B}{2} \right)+\sin 2C
\\
& \Rightarrow 2\cos \left( A+B \right)\sin \left( A-B \right)+\sin 2C \\
\end{align}$
Now, using the formula from equation (3) and equation (4) in the above equation. Then,
$\begin{align}
& 2\cos \left( A+B \right)\sin \left( A-B \right)+\sin 2C \\
& \Rightarrow -2\cos C\sin \left( A-B \right)+2\sin C\cos C \\
& \Rightarrow 2\cos C\left( \sin C-\sin \left( A-B \right) \right) \\
\end{align}$
Now, using the formula from equation (2) and equation (1) in the above equation. Then,
$\begin{align}
& 2\cos C\left( \sin C-\sin \left( A-B \right) \right) \\
& \Rightarrow 2\cos C\left( \sin \left( A+B \right)-\sin \left( A-B \right) \right) \\
& \Rightarrow 2\cos C\left( 2\cos \left( \dfrac{A+B+A-B}{2} \right)\sin \left(
\dfrac{A+B-A+B}{2} \right) \right) \\
& \Rightarrow 2\cos C\left( 2\cos A\sin B \right) \\
& \Rightarrow 4\cos A\sin B\cos C \\
\end{align}$
Now, from the above result, we can say that $\sin 2A-\sin 2B+\sin 2C=4\cos A\sin B\cos C$
Thus, $L.H.S=R.H.S$ .
Hence, proved.
Note: Here, the student should first understand what we have to prove in the question and then proceed in a stepwise manner while solving. We should also try to make use of trigonometric results like $\sin \left( \pi -\theta \right)=\sin \theta $ for making equations that will help us further in the solution. Moreover, the formula $\sin C-\sin D$ should be applied correctly with proper values and avoid making calculation mistakes while solving the problem.
Recently Updated Pages
The height of a solid metal cylinder is 20cm Its r-class-10-maths-ICSE

If a train crossed a pole at a speed of 60kmhr in 30 class 10 physics CBSE

Name the Writs that the High Courts are empowered to class 10 social science CBSE

A tower is 5sqrt 3 meter high Find the angle of el-class-10-maths-CBSE

Immediate cause of variations of A Mutations B Environmental class 10 biology CBSE

A rectangular container whose base is a square of side class 10 maths CBSE

Trending doubts
Why is there a time difference of about 5 hours between class 10 social science CBSE

Why is Sardar Vallabhbhai Patel called the Iron man class 10 social science CBSE

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

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Write an application to the principal requesting five class 10 english CBSE

What is the median of the first 10 natural numbers class 10 maths CBSE

