Solve for x sec(x)^2-2tan(x)=4

Problem

sec(x)−2*tan(x)=4

Solution

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

1+tan(x)−2*tan(x)=4

2. Rearrange the equation into a standard quadratic form by subtracting 4 from both sides.

tan(x)−2*tan(x)−3=0

3. Factor the quadratic expression by finding two numbers that multiply to −3 and add to −2

(tan(x)−3)*(tan(x)+1)=0

4. Set each factor to zero to find the possible values for tan(x)

tan(x)−3=0⇒tan(x)=3

tan(x)+1=0⇒tan(x)=−1

5. Solve for x by taking the arctangent of both sides, noting that the tangent function has a period of π

x=arctan(3)+n*π

x=arctan(−1)+n*π

6. Simplify the exact value for the second solution using arctan(−1)=−π/4

x=−π/4+n*π

Final Answer

sec(x)−2*tan(x)=4⇒x=arctan(3)+n*π,x=−π/4+n*π

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