The minimum positive period of the function y = 2sin2x is______ .

The minimum positive period of the function y = 2sin2x is______ .

0

0

f(x)=（(1/sin^4x）-1)(（1/cos^4）-1)
=(1-sin^4x)(1-cos^4x)/(sinxcosx)^4
=(sinxcosx)^2(1+sin^2x)(1+cos^2x)/(sinxcosx)^4
=(1+sin^2x)(1+cos^2x)/(sinx)^2(cosx)^2
=(1/sin^2x+1)(1/cos^2x+1)
=(1/sin^2x+1)(1/(1-sin^2x)+1)
Let sin ^ 2x = t
T>0
y=(1/t+1)(1/(1-t)+1)
It can be divided into a quadratic equation of one variable with respect to t
(y-1)t^2-t(y-1)+2=0
When y = 1, t has no solution
So t is not equal to 1
When y is not equal to 1:
△>=0
(y-1)^2-8(y-1)>=0
(y-1)(y-9）>=0
y>=9
Therefore, the minimum value of F (x) is 9

Find the minimum value of the function y = sin ^ 4x + cos ^ 4x, X (0, π / 6)

0

0

y=cos(π/3-x/2),x∈[-2π,2π]
=cos(x/2-π/3)
From 2K π - π ≤ X / 2 - π / 3 ≤ 2K π, K ∈ Z
2K π - 2 π / 3 ≤ X / 2 ≤ 2K π + π / 3, K ∈ Z is obtained
∴4kπ-4π/3≤x≤4kπ+2π/3,k∈Z
∵x∈[-2π,2π]
The increasing range is as follows:
[-4π/3,2π/3]

Find the function y = - cos (x 2−π 3) Monotonically increasing interval of

∵y=cos（x
2-π
3) The monotone decreasing interval of is y = - cos (x)
2-π
3) Monotonically increasing interval of,
From 2K π ≤ x
2-π
3 ≤ 2K π + π (K ∈ z), 2 π
3+4kπ≤x≤8π
3+4kπ（k∈Z），
The function y = - cos (x
2-π
3) The monotone increasing interval of the is [2 π]
3+4kπ，8π
3+4kπ]（k∈Z）．

Find the function y = - cos (x 2−π 3) Monotonically increasing interval of

∵y=cos（x
2-π
3) The monotone decreasing interval of is y = - cos (x)
2-π
3) Monotonically increasing interval of,
From 2K π ≤ x
2-π
3 ≤ 2K π + π (K ∈ z), 2 π
3+4kπ≤x≤8π
3+4kπ（k∈Z），
The function y = - cos (x
2-π
3) The monotone increasing interval of the is [2 π]
3+4kπ，8π
3+4kπ]（k∈Z）．

The function y = cos (π The interval of − x is increasing A. [2kπ-3π 4，2kπ+π 4]，k∈Z B. [2kπ-5π 4，2kπ−π 4]，k∈Z C. [2kπ+π 4，2kπ+5π 4]，k∈Z D. [2kπ-π 4，2kπ+3π 4]，k∈Z

According to the induction formula, y = cos (π)
4 − x) is y = cos (x − π)
4)，
Let - π + 2K π ≤ X - π
4 ≤ 2K π (K ∈ z), the solution is - 3 π
4+2kπ≤x≤π
4+2kπ（k∈Z），
The function y = cos (π)
The monotone increasing interval of 4 − x) is [- 3 π
4+2kπ，π
4+2kπ]（k∈Z）．
Therefore, a

Find the maximum and minimum of the period sum of F (x) = 2 and radical three sinxcosx + 2cos square X-1

F (x) = 2 and radical 3 sinxcosx + 2cos square X-1
=Radical three sin2x + cos2x
=2sin(2x+30°）
So the period is 2 π / 2 = π
The maximum value is 2 and the minimum value is - 2

It is proved that: (Sina + COSA) ^ 2 = 1 + 2Sin ^ 2acota

Left = sin? 2A + cos? A + 2sinacosa = 1 + 2sinacosa
Right = 1 + 2Sin? A * cosa / Sina = 1 + 2sinacosa = left
Proof of proposition

Simplification of sin (π / 4 + a) cosa sin (π / 4-A) Sina=

sin(π/4+a)cosa-sin(π/4-a)sina
=cos[π/2-(π/4+a)]cosa-sin(π/4-a)sina
=cos(π/4-a)cosa-sin(π/4-a)sina
=cos(π/4-a+a)
=√2/2