How do I find all solutions to:

tanx+3cotx=4?

The answer choices are:
a.) x=arctan3+nPi
b.) x=arccot3+nPi

actually there are other answers ...

tanx + 3cotx = 4
tanx + 3/tanx = 4

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

x = arctan 1 + npi or x = arctan3 + npi

Thanks! :D

To find all the solutions to the equation tan(x) + 3cot(x) = 4, we can use the following steps:

Step 1: Rewrite the equation in terms of sine and cosine.
Using the definitions of tangent and cotangent:

tan(x) = sin(x) / cos(x)
cot(x) = cos(x) / sin(x)

We can rewrite the equation as:

sin(x)/cos(x) + 3cos(x)/sin(x) = 4

Step 2: Multiply through by sin(x) * cos(x) to eliminate the denominators.

sin(x)*sin(x) + 3cos(x)*cos(x) = 4sin(x)*cos(x)

Simplifying further, we get:

sin^2(x) + 3cos^2(x) = 4sin(x)cos(x)

Step 3: Use the Pythagorean Identity for sine and cosine.

sin^2(x) + cos^2(x) = 1

Substitute this identity into the equation:

1 + 2cos^2(x) = 4sin(x)cos(x)

Step 4: Rearrange the equation.

2cos^2(x) - 4sin(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
cos(x) - 1 = 0

Solving these equations separately, we get:

cos(x) = 1/2
cos(x) = 1

Step 7: Take the inverse cosine (arccos) of both sides to solve for x.

For cos(x) = 1/2, the solutions are:

x = arccos(1/2) + 2nπ
x = π/3 + 2nπ
or
x = 5π/3 + 2nπ

For cos(x) = 1, the solutions are:

x = arccos(1) + 2nπ
x = 2nπ

Step 8: Finally, combine all possible values of x.

Combining the solutions, we have:

x = π/3 + 2nπ, 5π/3 + 2nπ, and 2nπ

Therefore, the correct answer choice is:

b.) x = arccot(3) + nπ

To find all solutions to the equation tan(x) + 3cot(x) = 4, we need to use trigonometric identities and algebraic manipulation. Let's break it down step by step:

1. Rewrite the equation using Pythagorean identities:
Recall that tan(x) = sin(x) / cos(x) and cot(x) = cos(x) / sin(x). Using these identities, we can rewrite the equation as:

sin(x)/cos(x) + 3cos(x)/sin(x) = 4

2. To simplify this expression, let's find the common denominator, which is cos(x) * sin(x). Multiply the first term (sin(x)/cos(x)) by sin(x)/sin(x) and the second term (3cos(x)/sin(x)) by cos(x)/cos(x):

sin^2(x) / (cos(x) * sin(x)) + 3cos^2(x) / (cos(x) * sin(x)) = 4

3. Combine the fractions by adding the numerators:

(sin^2(x) + 3cos^2(x)) / (cos(x) * sin(x)) = 4

4. Simplify the numerator of the fraction:
Using the identity sin^2(x) + cos^2(x) = 1, the above equation becomes:

(1 + 2cos^2(x)) / (cos(x) * sin(x)) = 4

5. Multiply both sides of the equation by (cos(x) * sin(x)) to clear the denominator:

1 + 2cos^2(x) = 4cos(x) * sin(x)

6. Rearrange the equation to form a quadratic equation:
Move all terms to one side of the equation:

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

7. Let's simplify this equation further:
Since 2cos^2(x) is common to both terms, we can factor it out:

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

8. We can recognize the expression in the parentheses as the square of a binomial:

2(cos(x) - sin(x)/√2)^2 = 0

9. Solve for the expression in parentheses:
Now we set the expression inside the square bracket equal to zero:

cos(x) - sin(x)/√2 = 0

10. Rearrange this equation to isolate cos(x):

cos(x) = sin(x)/√2

11. Divide both sides of the equation by sin(x):

cot(x) = 1/√2

12. Use the definition of the cotangent function:
Recall that cot(x) = 1/tan(x). So, we have:

1/tan(x) = 1/√2

13. Take the reciprocal of both sides:

tan(x) = √2

14. To find the solutions, we can take the inverse tangent of both sides:

x = arctan(√2)

15. Since tangent has a periodicity of π (180 degrees), we can add nπ to the solution to find all possible solutions:

x = arctan(√2) + nπ

So, the solution to the equation tan(x) + 3cot(x) = 4 is x = arctan(√2) + nπ.

Comparing this solution with the answer choices provided:
a.) x = arctan(3) + nπ is not the correct answer.
b.) x = arccot(3) + nπ is not the correct answer either.

Therefore, none of the answer choices match the correct solution.