Question

Question for Brainly Teachers​

Answer 1

Answer:

Let

$\theta + \phi = x \: \\ \theta - \phi =y \\ 2 \theta = x + y \: \: and \: \: 2 \phi = x - y$

It looks easy now

Open the determinate according row 3

$\sin(x + y) ( \sin(x) \sin(y) + \cos(x) \cos(y) ) \\ + \sin(x - y) ( \cos(x) \cos(y) - \sin(x) \sin(y) ) \\ \\ = \sin(x + y) \cos(x - y) + \sin(x - y) \cos(x + y) \\ \\ = \sin(x + y + x - y) \\ \\ = \sin(2x) \\ \\ = \sin(2( \theta + \phi))$

Answer 2

Let

$$\begin{lgathered}\theta + \phi = x \: \\ \theta - \phi =y \\ 2 \theta = x + y \: \: and \: \: 2 \phi = x - y\end{lgathered}$$

It looks easy now

Open the determinate according row 3

$$\begin{lgathered}\sin(x + y) ( \sin(x) \sin(y) + \cos(x) \cos(y) ) \\ + \sin(x - y) ( \cos(x) \cos(y) - \sin(x) \sin(y) ) \\ \\ = \sin(x + y) \cos(x - y) + \sin(x - y) \cos(x + y) \\ \\ = \sin(x + y + x - y) \\ \\ = \sin(2x) \\ \\ = \sin(2( \theta + \phi))\end{lgathered}$$