Question

How do you simplify $\cos \left( {x + \pi } \right)$ ?

Answer 1

Hint: In this question, we will use the basic formula of trigonometry that is the formula of $\cos \left( {A + B} \right)$ by which we obtain the simplest form of the given function and we use the value of $\cos \pi $ as $ - 1$ and the value of $\sin \pi $ as $0$.

Complete step by step solution:
In this question, we have given a trigonometric function that is $\cos \left( {x + \pi } \right)$, which needs to be simplified.
As we know that $\cos x$ is the trigonometric ratio which is the ratio of the length of base and the length of the hypotenuse of the right angle triangle. To simplify the given function, we will use the formula,
$ \Rightarrow \cos \left( {A + B} \right) = \cos A \cdot \cos B - \sin A \cdot \sin B$
We have given $\cos \left( {x + \pi } \right)$, so we will compare it with the formula and get,
$ \Rightarrow A = x$
$ \Rightarrow B = \pi $
Now we will substitute the obtained values in the above formula as,
$ \Rightarrow \cos \left( {x + \pi } \right) = \cos x \cdot \cos \pi - \sin x \cdot \sin \pi $
As we know that the value of $\cos \pi $ is $ - 1$ and the value of $\sin \pi $ is $0$, so now we will substitute it in the above expression as,
$ \Rightarrow \cos \left( {x + \pi } \right) = \cos x \cdot \left( { - 1} \right) - \sin x \cdot \left( 0 \right)$
Now, we will simplify the above expression as,
$ \Rightarrow \cos \left( {x + \pi } \right) = - \cos x - 0$
We will simplify it further and get,
$\therefore \cos \left( {x + \pi } \right) = - \cos x$

Therefore, the simplified value of the given function is $ - \cos x$.

Note:
If the function is in the form of $\cos \left( {x - \pi } \right)$ then we will be simplified it as,
we will use the formula,
$ \Rightarrow \cos \left( {A - B} \right) = \cos A \cdot \cos B + \sin A \cdot \sin B$
We have given $\cos \left( {x + \pi } \right)$, so we will compare it with the formula and get,
$ \Rightarrow A = x$
$ \Rightarrow B = \pi $
Now we will substitute the obtained values in the above formula as,
$ \Rightarrow \cos \left( {x - \pi } \right) = \cos x \cdot \cos \pi + \sin x \cdot \sin \pi $
As we know that the value of $\cos \pi $ is $ - 1$ and the value of $\sin \pi $ is $0$, so now we will substitute it in the above expression as,
$ \Rightarrow \cos \left( {x - \pi } \right) = \cos x \cdot \left( { - 1} \right) + \sin x \cdot \left( 0 \right)$
Now, we will simplify the above expression as,
$ \Rightarrow \cos \left( {x - \pi } \right) = - \cos x + 0$
We will simplify it further and get,
$\therefore \cos \left( {x - \pi } \right) = - \cos x$
Therefore, the simplified value of the given function is $ - \cos x$.
Hence, from the above simplification we can say that,
$\therefore \cos \left( {x - \pi } \right) = \cos \left( {x + \pi } \right) = - \cos x$.