Given the function f (x) = 1 + SiN x × cos X. (1) find the minimum positive period and monotone decreasing interval of F (x); (2) if Tan x = 2, find the value of F (x)

Given the function f (x) = 1 + SiN x × cos X. (1) find the minimum positive period and monotone decreasing interval of F (x); (2) if Tan x = 2, find the value of F (x)

one
f(x)=1+sinxcosx=1+(1/2)sin(2x)
Minimum positive period Tmin = 2 π / 2 = π
When 2K π + π / 2 ≤ 2x ≤ 2K π + 3 π / 2 (K ∈ z), f (x) decreases monotonically
kπ+π/4≤x≤kπ+3π/4 (k∈Z)
The monotone decreasing interval of the function is [K π + π / 4, K π + 3 π / 4] (K ∈ z)
two
tanx=sinx/cosx=2
sinx=2cosx
(sinx)^2+(cosx)^2=1
(2cosx)^2+(cosx)^2=1
5(cosx)^2=1
(cosx)^2=1/5
f(x)=1+sinxcosx=1+(2cosx)cosx=1+2(cosx)^2=1+2(1/5)=1+2/5=7/5

Is the axis of symmetry and the center of symmetry of the function f (a + x) = f (b-X) the same as that of the function f (a + x) = - f (b-X)

From F (a + x) = f (b-X), the image of function f (x) is axisymmetric
Axis of symmetry x = (a + X + b-X) / 2 = (a + b) / 2
From F (a + x) = - f (b-X), we can see that the image of function f (x) is a centrosymmetric figure
Symmetry center ((a + b) / 2,0)
Axisymmetry is different from centrosymmetry
Let's look at the definition
Axial symmetry:
When a figure is folded along a line, if the two sides of the line can overlap each other, the figure is called an axisymmetric figure, and the line is the axis of symmetry
If the image of a function is axisymmetric, the following conditions must be met:
x+x’=2a
y-y'=0
It's expressed as a function
f（x）=f（2a-x）
Centrosymmetry:
If a figure can be rotated 180 ° around a certain point and coincide with another figure, the two figures are said to be symmetrical or centrosymmetric about this point. This point is called the center of symmetry, and the corresponding point of the two figures is called the symmetrical point about the center
If the image of a function is centrosymmetric, the following conditions must be satisfied:
x+x‘=2a
y+y’=2b
It's expressed as a function
f（x）+f（2a-x）=2b

Given that x ≠ 0, f (x) satisfies f (x-1 / x) = x ^ 2 + 1 / x ^ 2, then the expression of F (x) is ()