Question

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

Answer 1

By using the basic trigonometric identity given below we can simplify the above expression that is $\cos (x - y) = \cos x\cos y + \sin x\sin y$ . We can prove the given statement by using $\cos (x + y) = \cos x\cos y - \sin x\sin y$ , by replacing $y$ by $- y$ . In order to solve and simplify the given expression we have to use the identity and express our given expression in the simplest form and thereby solve it.

Complete step-by-step solution:
To prove: $\cos (x - y) = \cos x\cos y + \sin x\sin y$
Proof:
We already know that $\cos (x + y) = \cos x\cos y - \sin x\sin y$ ,
Now using the expression given below ,
$\cos (x + y) = \cos x\cos y - \sin x\sin y$
Now, we have to replace $y$ by $- y$ to obtain the required result ,
We will get the following result ,
$\cos (x + ( - y)) = \cos x\cos ( - y) + \sin x\sin ( - y)$
As we know that $\cos ( - x) = \cos x$ and $\sin ( - x) = - \sin x$ ,
Therefore, we will get the following expression,
$\cos (x - y) = \cos x\cos y + \sin x\sin y$
Hence proved.

Thus we have proved that L.H.S = R.H.S i.e, $\cos (x - y) = \cos x\cos y + \sin x\sin y$

Note: Some other equations needed for solving these types of problem are:
$\cos (x + y) = \cos x\cos y - \sin x\sin y$ ,
$\cos ( - x) = \cos x$ ,
And
$\sin ( - x) = - \sin x$ .
Range of cosine and sine: $\left[ { - 1,1} \right]$ ,
We can prove the given statement by using $\cos (x + y) = \cos x\cos y - \sin x\sin y$ , by replacing $y$ by $- y$ . In order to solve and simplify the given expression we have to use the identity and express our given expression in the simplest form and thereby solve it.