2sec^2x -3tanx =5 for [0,2pi)

tan2(x) + 1 = sec2(x) identity

2(tan^2(x)+1)-3tanx=5
put that in standard form
au^2+bu+c=0 and solve for tan x

and what are the answers

To solve the trigonometric equation 2sec^2x - 3tanx = 5 for the given interval [0, 2π), we can follow these steps:

Step 1: Rewrite the equation in terms of sin and cos.
Since sec(x) is the reciprocal of cos(x) and tan(x) is the ratio of sin(x) and cos(x), we can rewrite the equation using these trigonometric functions:
2/cos^2(x) - 3(sin(x)/cos(x)) = 5

Step 2: Multiply through by cos^2(x) to eliminate the denominator:
2 - 3sin(x)cos(x) = 5cos^2(x)

Step 3: Rearrange the equation to bring all terms to one side:
5cos^2(x) + 3sin(x)cos(x) - 2 = 0

Step 4: Use a trigonometric identity to simplify the equation.
Using the double angle identity for cos(2x), we have:
5(1 - sin^2(x)) + 3sin(x)cos(x) - 2 = 0
5 - 5sin^2(x) + 3sin(x)cos(x) - 2 = 0
3sin(x)cos(x) - 5sin^2(x) + 3 = 0

Step 5: Factor out sin(x) from the equation:
sin(x)(3cos(x) - 5sin(x)) + 3 = 0

Step 6: Solve each part of the equation separately:
i) sin(x) = 0
From this, we get two solutions: x = π and x = 0.

ii) 3cos(x) - 5sin(x) = 0
To solve this trigonometric equation, we can divide both sides by cos(x) since cos(x) is not zero.

3 - 5tan(x) = 0
5tan(x) = 3
tan(x) = 3/5

To find the solutions in the given interval [0, 2π), we need to look for values of x where the sine function is zero and where the tangent function is equal to 3/5.

Therefore, the solutions for the equation 2sec^2x - 3tanx = 5 in the interval [0, 2π) are: x = 0, x = π, and the values of x where tan(x) = 3/5.