Solve for x 4sec(x)^2tan(x)=8tan(x)

Problem

4*sec2(x)*tan(x)=8*tan(x)

Solution

1. Subtract 8*tan(x) from both sides to set the equation to zero.

4*sec2(x)*tan(x)−8*tan(x)=0

2. Factor out the greatest common factor, which is 4*tan(x)

4*tan(x)*(sec2(x)−2)=0

3. Apply the zero product property by setting each factor equal to zero.

4*tan(x)=0

sec2(x)−2=0

4. Solve the first equation for x by finding where tan(x)=0

tan(x)=0

x=n*π

5. Solve the second equation for x by isolating sec(x)

sec2(x)=2

sec(x)=±√(,2)

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

cos(x)=±1/√(,2)

cos(x)=±√(,2)/2

7. Determine the values of x where cos(x)=±√(,2)/2

x=π/4+(n*π)/2

Final Answer

x=n*π,π/4+(n*π)/2

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