1)cos3x=1/22)1+√2 cosx/8=03)cos^2x-cosx sinx=04)2cos^2 x+4sin^2x=3

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1) To solve for x in the equation cos3x = 1/2, we need to find the values of x that satisfy the equation. Since cos3x = cos(3x), we can rewrite the equation as cos(3x) = 1/2.

Using the identity cos(3x) = 4cos^3x - 3cosx, we get:
4cos^3x - 3cosx = 1/2
4cos^3x - 3cosx - 1/2 = 0

Now, let cosx = y. We can rewrite the equation as:
4y^3 - 3y - 1/2 = 0

This is a cubic equation in y. We can solve for y using numerical methods or graph it to find the values of y that satisfy the equation. Then, we can find the corresponding values of x by taking the inverse cosine.

2) To solve the equation 1 + √2 cosx/8 = 0, we first isolate the cosine term:
√2 cosx/8 = -1
cosx/8 = -√2

Now, multiply both sides by 8 to get:
cosx = -8√2

This means that x can be any angle whose cosine is equal to -8√2.

3) To solve the equation cos^2x - cosx sinx = 0, we can factor out a common factor of cosx:
cosx(cosx - sinx) = 0

This equation is satisfied when either cosx = 0 or cosx - sinx = 0. Solving these two equations will give us the values of x that satisfy the original equation.

4) To solve the equation 2cos^2 x + 4sin^2x = 3, we use the Pythagorean identity sin^2x + cos^2x = 1 to rewrite the equation in terms of only one trigonometric function. Let's choose cosx:

2(1 - sin^2x) + 4sin^2x = 3
2 - 2sin^2x + 4sin^2x = 3
2 + 2sin^2x = 3
2sin^2x = 1
sin^2x = 1/2

Taking the square root gives us sinx = ±√(1/2) = ±1/√2
Since sinx can be ±1/√2, there are multiple values of x that satisfy the equation.