Solve for x 2sec(x)^2+tan(x)^2-3=0

Problem

2*sec(x)+tan(x)−3=0

Solution

1. Apply the Pythagorean identity to rewrite the equation in terms of a single trigonometric function using sec(x)=1+tan(x)

2*(1+tan(x))+tan(x)−3=0

2. Distribute the constant into the parentheses.

2+2*tan(x)+tan(x)−3=0

3. Combine like terms to simplify the equation.

3*tan(x)−1=0

4. Isolate the trigonometric term by adding 1 to both sides and dividing by 3.

3*tan(x)=1

tan(x)=1/3

5. Take the square root of both sides to solve for tan(x)

tan(x)=±√(,1/3)

tan(x)=±1/√(,3)

6. Find the general solution for x by identifying the angles where the tangent value is ±1/√(,3)

x=π/6+n*π

x=(5*π)/6+n*π

Final Answer

x=π/6+n*π,(5*π)/6+n*π

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