Solve for x sec(x)^2-sec(x)=2

Problem

sec2(x)−sec(x)=2

Solution

1. Rearrange the equation into a standard quadratic form by subtracting 2 from both sides.

sec2(x)−sec(x)−2=0

2. Substitute a variable to simplify the quadratic, letting u=sec(x)

u2−u−2=0

3. Factor the quadratic expression.

(u−2)*(u+1)=0

4. Solve for u by setting each factor to zero.

u=2

u=−1

5. Back-substitute sec(x) for u to find the values of x

sec(x)=2

sec(x)=−1

6. Convert the secant equations into cosine equations using the identity cos(x)=1/sec(x)

cos(x)=1/2

cos(x)=−1

7. Determine the general solutions for x based on the unit circle.

x=π/3+2*n*π

x=(5*π)/3+2*n*π

x=π+2*n*π

Final Answer

sec2(x)−sec(x)=2⇒x=π*(2*n+1),x=π/3+2*n*π,x=(5*π)/3+2*n*π

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