Which function has the same graph as y = 2cos(x + pi/4 )?

y = 2sin(x + pi/4 )
y = 2sin(x - pi/4 )
y = 2sin(x + 3pi/4 )
y = 2sin(x - 3pi/4 )

y = 2 sin (x + ) has the same graph of y = 2 cos (x + ) ⇒ 3rd answer

Step-by-step explanation:

Let us revise the rules of the trigonometric compound angles

cos(x + y) = cos x cos y - sin x sin y

sin(x + y) = sin x cos y + cos x sin y

Let us solve the problem using the rules above

y = 2 cos (x + )

∵ 2 cos (x + ) = 2[cos x cos  - sin x sin ]

∵ cos  =  and sin  =  

- Substitute them in the right hand side

∴ 2 cos (x + ) = 2[ cos x  -   sin x]

- Multiply the bracket in the right hand side by 2

∴ 2 cos (x + ) = cos x - sin x

∴ y = cos x - sin x

Now let us find the function which give the same right hand side of the function above

y = 2 sin (x + )

∵ 2 sin (x + ) = 2[sin x cos  + cos x sin ]

∵ sin  =  and cos  =  

- Substitute them in the right hand side

∴ 2 sin (x + ) = 2[ sin x  + cos x]

- Multiply the bracket in the right hand side by 2

∴ 2 sin (x + ) =   sin x  + cos x

- Switch the two terms of the right hand side

∴ 2 sin (x + ) =   cos x - sin x  

∴ y =   cos x - sin x

- The same with right hand side of the function above

∴ y = 2 sin (x + ) has the same graph of y = 2 cos (x + )