Question

How do you prove $\sin (x + y + z) = \sin x\cos y\cos z + \cos x\sin y\cos z + \cos x\cos y\sin z - \sin x\sin y\sin z$?

Answer 1

Hint: Here we have to prove the trigonometric function $\sin (x + y + z)$ is equal to $\sin x\cos y\cos z + \cos x\sin y\cos z + \cos x\cos y\sin z - \sin x\sin y\sin z$. In order to solve this question we first assume that $A = x + y$ and $B = z$ in the function $\sin (x + y + z)$ then we will use trigonometric identities such as $\sin (A + B) = \sin A\cos B + \cos A\sin B$ and $\cos (A + B) = \cos A\cos B - \sin A\sin B$ to get the required results

Complete step by step answer:
Here we have to prove the trigonometric function $\sin (x + y + z)$ is equal to
$\sin x\cos y\cos z + \cos x\sin y\cos z + \cos x\cos y\sin z - \sin x\sin y\sin z$
Now, consider the left side of the equation.
We have, $\sin (x + y + z)$
Let $A = x + y$ and $B = z$
So, we can write $\sin (x + y + z)$= $\sin (A + B)$

Using the identity $\sin (A + B) = \sin A\cos B + \cos A\sin B$.
Therefore, $\sin (x + y + z) = \sin (x + y)\cos z + \cos (x + y)\sin z$
Now, we will solve $\sin (x + y)$ and $\cos (x + y)$
Using the identities $\sin (A + B) = \sin A\cos B + \cos A\sin B$ and $\cos (A + B) = \cos A\cos B - \sin A\sin B$. We get,
$ \Rightarrow \sin (x + y) = \sin x\cos y + \cos x\sin y$
$ \Rightarrow \cos (x + y) = \cos x\cos y - \sin x\sin y$

Substituting these values in the function. we get,
$ \Rightarrow \sin (x + y + z) = (\sin x\cos y + \cos x\sin y)\cos z + (\cos x\cos y - \sin x\sin y)\sin z$
Solving the above equation. We get,
$ \Rightarrow \sin (x + y + z) = \sin x\cos y\cos z + \cos x\sin y\cos z + \cos x\cos y\sin z - \sin x\sin y\sin z$
Hence, the left side of the equation is equal to the right side of the equation. i.e.,
$\therefore \sin (x + y + z) = \sin x\cos y\cos z + \cos x\sin y\cos z + \cos x\cos y\sin z - \sin x\sin y\sin z$
Hence proved

Note: To solve these types of questions we have to split the angles first before using the identity. Here we can also split as $A = x$ and $B = y + z$ and after that use the trigonometric identities such as $\sin (A + B) = \sin A\cos B + \cos A\sin B$ and $\cos (A + B) = \cos A\cos B - \sin A\sin B$. Note that the addition and subtraction identities of $\cos $ have different signs in it. In addition, the subtraction sign is between the angles i.e., $\cos (A + B) = \cos A\cos B - \sin A\sin B$. Whereas in subtraction, the addition sign is between the angles i.e., $\cos (A - B) = \cos A\cos B + \sin A\sin B$. Some students are confused between trigonometric identities such as $\sin (A + B)$ and$\sin A + \sin B$. These both are different identities in one there is only a sum of angles and in second there is a sum of angles of $\sin $and similarly for $\cos A + \cos B$ and $\cos A - \cos B$.