Answer:
Option B is Correct.
Step-by-step explanation:
Given: [tex]y\,=\,3\,cos\,(2(x+\frac{\pi}{2}))-2[/tex]
To find: Equivalent Expression.
We use Following functions:
[tex]cos\,(\pi+x)=\,-cos\,x\:,\:cos\,(\frac{\pi}{2}+x)=\,-sin\,x\:\:and\:\:sin\,(\frac{\pi}{2}+x)=cos\,x[/tex]
First we simply the given expression,
[tex]y\,=\,3\,cos\,(2(x+\frac{\pi}{2}))-2[/tex]
[tex]y\,=\,3\,cos\,(2x+2\frac{\pi}{2}))-2[/tex]
[tex]y\,=\,3\,cos\,(2x+\pi)-2[/tex]
[tex]y\,=\,3\,cos\,(\pi+2x)-2[/tex]
[tex]y\,=\,3\,(-cos\,2x)-2[/tex] ( using above mentioned result )
[tex]y\,=\,-3\,cos\,2x-2[/tex] ............................(1)
Option A:
[tex]y\,=\,3\,sin\,(2(x+\frac{\pi}{4}))-2[/tex]
[tex]y\,=\,3\,sin\,(2x+2\frac{\pi}{4}))-2[/tex]
[tex]y\,=\,3\,sin\,(2x+\frac{\pi}{2})-2[/tex]
[tex]y\,=\,3\,sin\,(\frac{\pi}{2}+2x)-2[/tex]
[tex]y\,=\,3\,cos\,2x-2[/tex] ( using above mentioned result )
Since, it is not equal to (1)
Therefore, It is Not correct Option.
Option B:
[tex]y\,=\,-3\,sin\,(2(x+\frac{\pi}{4}))-2[/tex]
[tex]y\,=\,-3\,sin\,(2x+2\frac{\pi}{4}))-2[/tex]
[tex]y\,=\,-3\,sin\,(2x+\frac{\pi}{2})-2[/tex]
[tex]y\,=\,-3\,sin\,(\frac{\pi}{2}+2x)-2[/tex]
[tex]y\,=\,-3\,cos\,2x-2[/tex] ( using above mentioned result )
Since, it is equal to (1)
Therefore, It is Correct Option.
Option C:
[tex]y\,=\,3\,cos\,(2(x+\frac{\pi}{4}))-2[/tex]
[tex]y\,=\,3\,cos\,(2x+2\frac{\pi}{4}))-2[/tex]
[tex]y\,=\,3\,cos\,(2x+\frac{\pi}{2})-2[/tex]
[tex]y\,=\,3\,cos\,(\frac{\pi}{2}+2x)-2[/tex]
[tex]y\,=\,3\,(-sin\,2x)-2[/tex] ( using above mentioned result )
[tex]y\,=\,-3\,sin\,2x-2[/tex]
Since, it is not equal to (1)
Therefore, It is Not correct Option.
Option D:
[tex]y\,=\,-3\,cos\,(2(x+\frac{\pi}{2}))-2[/tex]
[tex]y\,=\,-3\,cos\,(2x+2\frac{\pi}{2}))-2[/tex]
[tex]y\,=\,-3\,cos\,(2x+\pi)-2[/tex]
[tex]y\,=\,-3\,cos\,(\pi+2x)-2[/tex]
[tex]y\,=\,-3\,(-cos\,2x)-2[/tex] ( using above mentioned result )
[tex]y\,=\,3\,cos\,2x-2[/tex]
Since, it is not equal to (1)
Therefore, It is Not correct Option.
Therefore, Option B is Correct.
