Question

$\cos \alpha .\sin (\beta -\gamma )+\cos \beta .\sin (\gamma -\alpha )+\cos \gamma .\sin (\alpha -\beta )$ is equal to
A . 0
B . $\dfrac{1}{2}$
C . 1
D . $\cos \alpha \cos \beta \cos \gamma $

Answer 1

Hint: In the given question we have to find the value of $\cos \alpha .\sin (\beta -\gamma )+\cos \beta .\sin (\gamma -\alpha )+\cos \gamma .\sin (\alpha -\beta )$. As we see an identity of sin is used in this question. So first we expand the equation by using the trigonometric identity of $\sin (x-y)$. Then by opening all the brackets and adding and subtracting the terms, we are able to get the desired answer and choose the correct option.

Formula Used:
In this question, we use the trigonometric identity which is described as below:-
$\sin (x-y)=\sin x\cos y-\cos x\sin y$

Complete step- by- step Solution:
Given that $\cos \alpha .\sin (\beta -\gamma )+\cos \beta .\sin (\gamma -\alpha )+\cos \gamma .\sin (\alpha -\beta )$……………………..(1)
We know the trigonometric identity
$\sin (x-y)=\sin x\cos y-\cos x\sin y$
then $\sin (\beta -\gamma )=(\sin \beta \cos \gamma -\sin \gamma \cos \beta )$
and $\sin (\gamma -\alpha )=(\sin \gamma \cos \alpha -\sin \alpha \cos \gamma )$
and $\sin (\alpha -\beta )=(\sin \alpha \cos \beta -\sin \beta \cos \alpha )$
Put the above identity in equation (1), we get
$\cos \alpha (\sin \beta \cos \gamma -\sin \gamma \cos \beta )+\cos \beta (sin\gamma \cos \alpha -\sin \alpha \cos \gamma )+\cos \gamma (\sin \alpha \cos \beta -\sin \beta \cos \alpha )$
Now by opening the brackets of the above equation, we get
$\cos \alpha \sin \beta \cos \gamma -\cos \alpha \sin \gamma \cos \beta +\cos \beta sin\gamma \cos \alpha -\cos \beta \sin \alpha \cos \gamma +\cos \gamma \sin \alpha \cos \beta -\cos \gamma \sin \beta \cos \alpha $
We see in the above equations, there are two similar terms with opposite signs. So they cancel each other.
Therefore, by cancelling all the similar terms with opposite signs, we get
$\cos \alpha .\sin (\beta -\gamma )+\cos \beta .\sin (\gamma -\alpha )+\cos \gamma .\sin (\alpha -\beta )$ = 0

Thus, Option (A) is correct.

Note: In these type of questions, students made mistake that they started solving the whole equation at one time. By solving the equation and putting the identities at one time makes us confused and we are not able to solve the question completely or we take extra time. By solving the equations in small parts and then combining them makes the question easy to solve and we solve it in lesser time.