00:01
Hi there, it is given in the question that cosine of x plus pi by 3 plus cosine of x minus pi by 4 is equal to 0.
00:10
Now we are asked to find the value of x.
00:14
We will convert the equation to the form cosine of a plus b plus cosine of a minus b is equal to 0.
00:22
We have the result that cosine of a plus b plus cosine of a minus b is equal to 2 multiplied by cosine a cosine b.
00:37
So this will be the result which we are going to use.
00:42
For that, let x plus a plus b is equal to x plus pi by 3 and x plus a minus 3.
00:54
Minus b equal to x minus pi by 4.
01:01
So here we need to solve these equations.
01:04
From here we'll be getting that.
01:06
This implies that a plus b is equal to pi by 3 and a minus b is equal to negative pi by 4.
01:18
Let us name this as equation number 1 and name this as equation number 2.
01:23
So the operation that is equation number 1.
01:28
1 minus equation number 2 implies from here we'll be getting that.
01:33
2b is equal to 7 pi divided by 12 and in turn we'll be getting b is equal to 7 pi divided by 24.
01:47
Now substitute b is equal to 7 pi divided by 24 in the equation number 1.
01:53
So equation number 1 was a plus b is equal to pi divided by 3.
01:58
So we'll be getting this as a plus 7 pi divided by 24 is equal to pi divided by 3.
02:08
And this implies a is equal to pi divided by 3 minus 7 pi divided by 24.
02:17
And that is equal to pi divided by 24.
02:22
So we have x plus pi divided by 3 is equal to x plus pi divided by 3 is equal to x plus pi divided by 24.
02:29
Divided by 24 plus 7 pi divided by 24 and x minus pi divided by 4 is equal to x plus pi divided by 24 minus 7 pi divided by 24.
02:51
So we can write cosine of x plus pi divided by 3 plus cosine of x minus pi divided by 4 is equal to cosine of x plus pi divided by 24 plus 7 pi divided by 24 plus cosine of x plus pi divided by 24 minus 7 pi divided by 24.
03:15
We already had the equation that is cos sine of a plus b plus cosine of a minus b is equal to 2 multiplied by cosine a multiplied by cosine by cosine b multiplied by cosine b.
03:31
So using that, here we can see that a, in place of a we have, in place of a, we have x plus pi divided by 24...