The graph of the function is
- A.a straight line passing through with slope
- B.a straight line passing through
- C.a parabola with vertex
- D.a straight line passing through the point and parallel to the -axis
Show answer and solution
Correct option: D
Solution
Using the identity: cos x · cos(x+2) - cos²(x+1)
Write cos x = cos((x+1)-1) and cos(x+2) = cos((x+1)+1), then:
cos x · cos(x+2) = [cos(x+1)cos1 + sin(x+1)sin1][cos(x+1)cos1 - sin(x+1)sin1]
= cos²(x+1)cos²1 - sin²(x+1)sin²1
So the expression = cos²(x+1)cos²1 - sin²(x+1)sin²1 - cos²(x+1)
= cos²(x+1)(cos²1 - 1) - sin²(x+1)sin²1
= -cos²(x+1)sin²1 - sin²(x+1)sin²1
= -sin²1[cos²(x+1) + sin²(x+1)]
= -sin²1
This is a constant function, so the graph is a horizontal straight line (parallel to X-axis) at y = -sin²1. Since it holds for all x, it passes through (π/2, -sin²1), confirming option 4.
Write cos x = cos((x+1)-1) and cos(x+2) = cos((x+1)+1), then:
cos x · cos(x+2) = [cos(x+1)cos1 + sin(x+1)sin1][cos(x+1)cos1 - sin(x+1)sin1]
= cos²(x+1)cos²1 - sin²(x+1)sin²1
So the expression = cos²(x+1)cos²1 - sin²(x+1)sin²1 - cos²(x+1)
= cos²(x+1)(cos²1 - 1) - sin²(x+1)sin²1
= -cos²(x+1)sin²1 - sin²(x+1)sin²1
= -sin²1[cos²(x+1) + sin²(x+1)]
= -sin²1
This is a constant function, so the graph is a horizontal straight line (parallel to X-axis) at y = -sin²1. Since it holds for all x, it passes through (π/2, -sin²1), confirming option 4.
[1997]