Solve:

sec(x)^2= 3tanx +1

*** It is suppose to be the sqaure root of 3 but i could find the symbol so just wrote the 3 without it but it is suppose to be the square root of 3 times tanx + 1

Rewrite this using the trig identity

sec^2(x) = 1 + tan^2(x)

Then treat tanx as a new variable, y

1 + tan^2x = sqrt3*(tanx + 1)

1 + y^2 = sqrt3*(y + 1)

y^2 -sqrt3*y -(sqrt3 -1) = 0

Solve the quadratic equation for y and then use x = arctan y to solve for x

To solve the equation sec(x)^2 = √3tan(x) + 1, we can simplify the equation and then use trigonometric identities to rewrite it in terms of a single trigonometric function.

1. Start by recognizing that sec(x)^2 is the reciprocal of cos(x)^2: sec(x)^2 = 1/cos(x)^2.

2. Replace sec(x)^2 in the equation with 1/cos(x)^2: 1/cos(x)^2 = √3tan(x) + 1.

3. Multiply both sides of the equation by cos(x)^2 to eliminate the fraction: 1 = (√3tan(x) + 1)cos(x)^2.

4. Distribute cos(x)^2 to both terms on the right side: 1 = √3tan(x)cos(x)^2 + cos(x)^2.

5. Recognize that tan(x) = sin(x)/cos(x) and simplify the equation further: 1 = √3(sin(x)/cos(x))cos(x)^2 + cos(x)^2.

6. Cancel out cos(x) terms on the right side: 1 = √3sin(x)cos(x) + cos(x)^2.

7. Rearrange the equation: 1 - cos(x)^2 = √3sin(x)cos(x).

8. Apply the identity sin^2(x) + cos^2(x) = 1: sin^2(x) = 1 - cos^2(x).

9. Substitute sin^2(x) into the equation: 1 - cos(x)^2 = √3sin(x)cos(x).

10. Replace 1 - cos(x)^2 with sin^2(x): sin^2(x) = √3sin(x)cos(x).

11. Divide both sides of the equation by sin(x): sin(x) = √3cos(x).

12. Divide both sides of the equation by cos(x): tan(x) = √3.

Therefore, the equation sec(x)^2 = √3tan(x) + 1 is satisfied when tan(x) = √3.