Question

State whether the following are true or false.
$\sin (A+B)=\sin A+\sin B$

• A. True
• B. False

Answer 1

The correct option is B False

We know the identity $\sin (A+B)=\sin A.\cos B+\cos .A\sin B$

For example lets take $A={45}^0$ and $B={45}^0$

Thus $\sin ({45}^0+{45}^0)=\sin {45}^0.\cos {45}^0+\cos {45}^0\sin {45}^0$

$\Rightarrow \sin {90}^0=\frac{1}{\sqrt{2}}.\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}.\frac{1}{\sqrt{2}}$

$\Rightarrow 1=\frac{1}{2}+\frac{1}{2}$

$\Rightarrow 1=1$ which is true.

Now take $\sin (A+B)=\sin A+\sin B$

$\Rightarrow \sin ({45}^0+{45}^0)=\sin {45}^0+\sin {45}^0$

$\Rightarrow \sin {90}^0=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}$

$\Rightarrow 1\ne \frac{2}{\sqrt{2}}$

Hence, given statement is false.