VERY URGENT! PLEASE HELP SOLVE!

tan(x)+1=sqrt(3)+sqrt(3)cot(x)as well as

sin(2x)= 2 cos^(x)

4cos2(x) = 8sin(x)cos(X)

1)sin(x)/cos(x)+1=sqrt(3)(1+cos(x)/sin(x))

(sin(x)+cos(x))/cos(x)=
=sqrt(3)(sin(x)+cos(x))/sin(x)
1/cos(x)=sqrt(3)/sin(x)
tan(x)=sqrt(3)
x=Pi/3+Pi*n

2)cos^(x)---???

3)cos2(x)=cos(2x) or cos^2(x)?

To solve the equations:

1) tan(x) + 1 = sqrt(3) + sqrt(3)cot(x)

Let's simplify this equation step by step:

Step 1: Rewrite cot(x) as 1/tan(x) using the reciprocal property.

tan(x) + 1 = sqrt(3) + sqrt(3)(1/tan(x))

Step 2: Multiply both sides of the equation by tan(x) to eliminate the denominator:

tan(x)(tan(x) + 1) = tan(x)(sqrt(3) + sqrt(3)(1/tan(x)))

tan^2(x) + tan(x) = sqrt(3)tan(x) + sqrt(3)

Step 3: Rearrange the terms:

tan^2(x) - sqrt(3)tan(x) + tan(x) - sqrt(3) = 0

Step 4: Combine like terms:

tan^2(x) - sqrt(3)tan(x) + (tan(x) - sqrt(3)) = 0

Step 5: Group the terms:

(tan^2(x) + tan(x)) - sqrt(3)(tan(x) - sqrt(3)) = 0

Step 6: Factor the trinomial:

(tan(x) + 1)(tan(x) - sqrt(3)) - sqrt(3)(tan(x) - sqrt(3)) = 0

Step 7: Simplify further:

(tan(x) + 1 - sqrt(3))(tan(x) - sqrt(3)) = 0

Step 8: Set each factor equal to zero and solve for x:

tan(x) + 1 - sqrt(3) = 0 OR tan(x) - sqrt(3) = 0

For the first equation, subtract 1 and sqrt(3) from both sides:

tan(x) = -1 + sqrt(3)

To find x, take the inverse tangent (arctan) of both sides using a calculator:

x = arctan(-1 + sqrt(3))

For the second equation, add sqrt(3) to both sides:

tan(x) = sqrt(3)

Again, take the inverse tangent of both sides using a calculator:

x = arctan(sqrt(3))

Therefore, the solutions are:
x = arctan(-1 + sqrt(3))
x = arctan(sqrt(3))

2) sin(2x) = 2cos^2(x)

Step 1: Recall the double-angle identity for sine:

sin(2x) = 2sin(x)cos(x)

Step 2: Substitute 2sin(x)cos(x) for sin(2x):

2sin(x)cos(x) = 2cos^2(x)

Step 3: Cancel out the common factor of 2:

sin(x)cos(x) = cos^2(x)

Step 4: Divide both sides by cos(x) to eliminate it:

sin(x) = cos(x)

Step 5: To solve this equation, either we can use the identities or examine the unit circle values.

Considering the unit circle, when sin(x) = cos(x), we have two possible solutions:

x = 45 degrees + n * 90 degrees or x = 225 degrees + n * 90 degrees

where n is an integer.

3) 4cos(2x) = 8sin(x)cos(x)

Step 1: Use the double-angle identity for cosine:

4(2cos^2(x) - 1) = 8sin(x)cos(x)

Step 2: Distribute the 4:

8cos^2(x) - 4 = 8sin(x)cos(x)

Step 3: Rearrange the terms:

8cos^2(x) - 8sin(x)cos(x) - 4 = 0

Step 4: Divide the whole equation by 4 to simplify it:

2cos^2(x) - 2sin(x)cos(x) - 1 = 0

Step 5: Factor the quadratic equation:

(2cos(x) - 1)(cos(x) - 1) = 0

Step 6: Set each factor equal to zero and solve for x:

2cos(x) - 1 = 0 or cos(x) - 1 = 0

For the first equation, add 1 and divide by 2:

cos(x) = 1/2

To find x, take the inverse cosine (arccos) of both sides using a calculator:

x = arccos(1/2)

For the second equation:

cos(x) = 1

Again, take the inverse cosine of both sides using a calculator:

x = arccos(1)

Therefore, the solutions are:
x = arccos(1/2)
x = arccos(1)