What is the general solution for sec^2(x)+tan^2(x)=5tan(x) ?

The general solution for sec^2(x)+tan^2(x)=5tan(x) is x=1.15759…+pin,x=0.21580…+pin

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sec^2(x)+tan^2(x)=5tan(x)

sec^{2} (x) + tan^{2} (x) = 5 tan (x)

x = 1.15759 \dots + πn, x = 0.21580 \dots + πn

Solution steps

sec^{2} (x) + tan^{2} (x) = 5 tan (x)

Subtract

5 tan (x)

from both sides

sec^{2} (x) + tan^{2} (x) - 5 tan (x) = 0

Rewrite using trig identities

1 + 2 tan^{2} (x) - 5 tan (x) = 0

Solve by substitution

tan (x) = \frac{5 + 17}{4}, tan (x) = \frac{5 - 17}{4}

tan (x) = \frac{5 + 17}{4} : x = arctan (\frac{5 + 17}{4}) + πn

tan (x) = \frac{5 - 17}{4} : x = arctan (\frac{5 - 17}{4}) + πn

Combine all the solutions

x = arctan (\frac{5 + 17}{4}) + πn, x = arctan (\frac{5 - 17}{4}) + πn

Show solutions in decimal form

x = 1.15759 \dots + πn, x = 0.21580 \dots + πn

sin (x) = \frac{9}{12}

- 2 cos (x) + 3 sin (x) = 2

cos (x + \frac{π}{6}) - 1 = cos (x - \frac{π}{6})

cos (2 x) cos (x) - sin (2 x) sin (x) = \frac{- 3}{2}

2 sin (θ) = cos (2 θ)

What is the general solution for sec^2(x)+tan^2(x)=5tan(x) ?
The general solution for sec^2(x)+tan^2(x)=5tan(x) is x=1.15759…+pin,x=0.21580…+pin