Solve for x sec(x)^2=4

Problem

sec2(x)=4

Solution

1. Take the square root of both sides of the equation to isolate the secant function.

sec(x)=±√(,4)

sec(x)=±2

2. Rewrite the equation using the reciprocal identity sec(x)=1/cos(x) to express the equation in terms of cosine.

1/cos(x)=±2

cos(x)=±1/2

3. Identify the reference angle for which the cosine value is 1/2 in the first quadrant.

x=π/3

4. Determine the general solutions by considering all quadrants where the cosine is 1/2 or −1/2 and accounting for the period of the cosine function, which is 2*π

x=π/3+n*π

x=(2*π)/3+n*π

5. Combine the solutions into a single expression where n is any integer.

x=π/3+n*π

x=−π/3+n*π

Final Answer

sec2(x)=4⇒x=π/3+n*π,(2*π)/3+n*π

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