Question

$\sin (\text{A}+\text{B}+\text{C})$ is equal to

• A. $\sin \text{A}\cos \text{B}\cos \text{C}[\tan \text{A}+\tan \text{B}+\tan \text{C}-\tan \text{A}\tan \text{B}\tan \text{C}]$
  
• B. $\cos \text{A}\cos \text{B}\cos \text{C}[\tan \text{A}+\tan \text{B}+\tan \text{C}+\tan \text{A}\tan \text{B}\tan \text{C}]$
  
• C. $\cos \text{A}\cos \text{B}\cos \text{C}[\tan \text{A}+\tan \text{B}+\tan \text{C}-\tan \text{A}\tan \text{B}\tan \text{C}]$
  
• D. $\cos \text{A}\sin \text{B}\cos \text{C}[\tan \text{A}+\tan \text{B}+\tan \text{C}-\tan \text{A}\tan \text{B}\tan \text{C}]$
  

Answer 1

The correct option is C $\cos \text{A}\cos \text{B}\cos \text{C}[\tan \text{A}+\tan \text{B}+\tan \text{C}-\tan \text{A}\tan \text{B}\tan \text{C}]$


Given : $\sin (\text{A}+\text{B}+\text{C})$
$\sin (\text{A}+\text{B}+\text{C})=\sin (\text{A}+(\text{B}+\text{C}\left)\right)$
We know, $\sin (A+B)=\sin A\cos B+\cos A\sin B$
$\Rightarrow \sin (\text{A}+\text{B}+\text{C})=\sin \text{A}\cos (\text{B}+\text{C})+\cos \text{A}\sin (\text{B}+\text{C})$

As, $\cos (A+B)=\cos A\cos B-\sin A\sin B$
$\Rightarrow \sin (\text{A}+\text{B}+\text{C})=\sin \text{A}[\cos \text{B}\cos \text{C}-\sin \text{B}\sin \text{C}]+\cos \text{A}[\sin \text{B}\cos \text{C}+\cos \text{B}\sin \text{C}]$

$\Rightarrow \sin (\text{A}+\text{B}+\text{C})=\sin \text{A}\cos \text{B}\cos \text{C}-\sin \text{A}\sin \text{B}\sin \text{C}+\cos \text{A}\sin \text{B}\cos \text{C}+\cos \text{A}\cos \text{B}\sin \text{C}$

Multiply and divide by $\cos \text{A}\cos \text{B}\cos \text{C}$ on R.H.S.

$\Rightarrow \sin (\text{A}+\text{B}+\text{C})=\cos \text{A}\cos \text{B}\cos \text{C}\left[\frac{\sin \text{A}\cos \text{B}\cos \text{C}-\sin \text{A}\sin \text{B}\sin \text{C}+\cos \text{A}\sin \text{B}\cos \text{C}+\cos \text{A}\cos \text{B}\sin \text{C}}{\cos \text{A}\cos \text{B}\cos \text{C}}\right]$

$\Rightarrow \sin (\text{A}+\text{B}+\text{C})=\cos \text{A}\cos \text{B}\cos \text{C}[\tan \text{A}-\tan \text{A}\tan \text{B}\tan \text{C}+\tan \text{B}+\tan \text{C}]$

$\Rightarrow \sin (\text{A}+\text{B}+\text{C})=\cos \text{A}\cos \text{B}\cos \text{C}[\tan \text{A}+\tan \text{B}+\tan \text{C}-\tan \text{A}\tan \text{B}\tan \text{C}]$