1. 4sin^2x=3tan^2x-1

2. 8cos^2x-4cos^4x=3

3. 3secx-cosx=2

Solve for values of x where 0 less/equal to x or less than 2pi. Express all answers in solution set form.

1.

4sin^2x = 3sin^2x/cos^2x - 1
multiply by cos^2x
4sin^2xcos^2x = 3sin^2x - cos^2x
4sin^2xcos^2x = 3sin^2 - (1 - sin^2x)
4sin^2xcos^2x = 4sin^2 - 1
4sin^2xcos^2x - 4sin^2 = - 1
4sin^2x(cos^2 - 1) = -1
4sin^2x(-sin^2x) = -1
-4sin^4x = -1
sin^4x = 1/4
sin^2 = ± 1/2
sinx = ±1/√2
x = π/4 , 3π/4 , 5π/4, and 7π/4

2.
let cos^2x = y, then we have
8y - 4y^2 = 3
4y^2 - 8y + 3 = 0
(2y - 1)(2y - 3) = 0
y = 1/2 or y = 3/2
so cos^2x = 1/2 or cos^2x = 3/2
cosx = ±1/√2 or cosx = ±√(3/2)
the second part is not possible since -1≤cosx≤+1
so cosx = ±1/√2
x = π/4 , 3π/4 , 5π/4, and 7π/4

try the third yourself, it is quite easy.

Step 1: Solve the equation 4sin^2x = 3tan^2x - 1

To simplify the equation, we can use the identity tan^2x = sec^2x - 1. We substitute this into the equation:

4sin^2x = 3(sec^2x - 1) - 1

Step 2: Expand the equation

4sin^2x = 3sec^2x - 3 - 1

4sin^2x = 3sec^2x - 4

Step 3: Substitute sin^2x with 1 - cos^2x

4(1 - cos^2x) = 3sec^2x - 4

Step 4: Simplify the equation

4 - 4cos^2x = 3sec^2x - 4

Step 5: Move all terms to one side to get a quadratic equation

4cos^2x - 3sec^2x + 4 = 0

Step 6: Solve the quadratic equation for cos^2x

We can use the quadratic formula to solve for cos^2x, considering it as a variable.

Quadratic formula: cos^2x = [-b ± √(b^2 - 4ac)] / 2a

a = 4, b = 0, c = -3

cos^2x = [0 ± √(0 - 4(4)(-3))] / (2 * 4)

cos^2x = ±√(-48) / 8

cos^2x = ±√(-6)

Since there is no real value for the square root of -6, this equation has no real solutions.

Therefore, the equation 4sin^2x = 3tan^2x - 1 has no real solutions.

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Step 1: Solve the equation 8cos^2x - 4cos^4x = 3

Step 2: This equation can be simplified by factoring out a common term.

8cos^2x - 4cos^4x - 3 = 0

Step 3: We can now factor this equation:

(2cos^2x - 1)(4cos^2x + 3) = 0

Step 4: Set each factor equal to zero and solve for cos^2x:

2cos^2x - 1 = 0 or 4cos^2x + 3 = 0

Step 5: Solve for cos^2x in the first equation:

2cos^2x - 1 = 0

2cos^2x = 1

cos^2x = 1/2

Step 6: Take the square root of both sides:

cosx = ±√(1/2)

cosx = ±√(2/2)

cosx = ±√(2)/2

This gives us two possible solutions: cosx = √(2)/2 or cosx = -√(2)/2.

So, the solutions for the equation 8cos^2x - 4cos^4x = 3 are x = π/4 + nπ/2 or x = 3π/4 + nπ/2, where n is an integer.

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Step 1: Solve the equation 3secx - cosx = 2

Step 2: Rewrite the equation using the identity secx = 1/cosx:

3(1/cosx) - cosx = 2

Step 3: Multiply through by cosx to eliminate fractions:

3 - cos^2x = 2cosx

Step 4: Rearrange the equation:

cos^2x + 2cosx - 3 = 0

Step 5: Factor the quadratic equation:

(cosx + 3)(cosx - 1) = 0

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

cosx + 3 = 0 or cosx - 1 = 0

cosx = -3 or cosx = 1

Step 7: Check for extraneous solutions:

cosx = -3 is not a valid solution since the cosine function only takes values between -1 and 1.

cosx = 1 is a valid solution.

Therefore, the solution for the equation 3secx - cosx = 2 is x = 2nπ, where n is an integer.

To solve these trigonometric equations, we'll use some basic trigonometric identities and algebraic techniques. Here's how to approach each of the equations:

1. 4sin^2x = 3tan^2x - 1:

Step 1: Replace tan^2x with sin^2x/cos^2x using the identity tan^2x = sin^2x / cos^2x.

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

Step 2: Multiply both sides of the equation by cos^2x to eliminate the fraction:

4sin^2x * cos^2x = 3sin^2x - cos^2x

Step 3: Use the Pythagorean identity sin^2x + cos^2x = 1 to simplify the equation:

4(1 - cos^2x) * cos^2x = 3sin^2x - cos^2x

Step 4: Distribute and collect like terms:

4cos^2x - 4cos^4x = 3sin^2x - cos^2x

Step 5: Move all terms to one side of the equation:

4cos^2x - 4cos^4x - 3sin^2x + cos^2x = 0

Step 6: Combine like terms:

5cos^2x - 4cos^4x - 3sin^2x = 0

Now you have a quadratic equation in terms of cos^2x. Solve it using algebraic techniques, such as factoring, completing the square, or the quadratic formula. Once you find the solutions for cos^2x, you can determine the values of x by taking the inverse cosine (arccos) of those solutions.

2. 8cos^2x - 4cos^4x = 3:

Step 1: Set the equation equal to 0:

8cos^2x - 4cos^4x - 3 = 0

Step 2: Rearrange the terms:

-4cos^4x + 8cos^2x - 3 = 0

Step 3: Factor out a common factor, if possible:

(2cos^2x - 1)(-2cos^2x + 3) = 0

Step 4: Set each factor equal to 0 and solve for cos^2x:

2cos^2x - 1 = 0, or -2cos^2x + 3 = 0

Solving these equations will give you the values of cos^2x. Take the inverse cosine (arccos) to find the values of x.

3. 3secx - cosx = 2:

Step 1: Rewrite secx in terms of cosine using the identity secx = 1/cosx:

3(1/cosx) - cosx = 2

Step 2: Multiply both sides of the equation by cosx to eliminate the fraction:

3 - cos^2x = 2cosx

Step 3: Rearrange the equation:

cos^2x + 2cosx - 3 = 0

Step 4: Factor the quadratic equation:

(cosx + 3)(cosx - 1) = 0

Step 5: Set each factor equal to 0 and solve for cosx:

cosx + 3 = 0, or cosx - 1 = 0

Solve these equations to find the values of cosx. Again, take the inverse cosine (arccos) to determine the values of x corresponding to those solutions.