What is the general solution for tan(x)sec(x)=-2tan(x) ?

The general solution for tan(x)sec(x)=-2tan(x) is x=pin,x=(2pi)/3+2pin,x=(4pi)/3+2pin

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tan(x)sec(x)=-2tan(x)

tan (x) sec (x) = - 2 tan (x)

x = πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

tan (x) sec (x) = - 2 tan (x)

Subtract

- 2 tan (x)

from both sides

tan (x) sec (x) + 2 tan (x) = 0

Factor

tan (x) sec (x) + 2 tan (x) : tan (x) (sec (x) + 2)

tan (x) sec (x) + 2 tan (x)

Factor out common term

tan (x)

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

tan (x) (sec (x) + 2) = 0

Solving each part separately

tan (x) = 0 or sec (x) + 2 = 0

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

sec (x) + 2 = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

sec (x) + 2 = 0

Move

2

to the right side

sec (x) + 2 = 0

Subtract

2

from both sides

sec (x) + 2 - 2 = 0 - 2

Simplify

sec (x) = - 2

sec (x) = - 2

General solutions for

sec (x) = - 2

sec (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Combine all the solutions

x = πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

1 + sin^{2} (x) = 2

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

cos (x) tan (x) = 0

cos^{2} (x) + 4 cos (x) + 3 = 0

sin (x) + sin^{2} (x) = 0

What is the general solution for tan(x)sec(x)=-2tan(x) ?
The general solution for tan(x)sec(x)=-2tan(x) is x=pin,x=(2pi)/3+2pin,x=(4pi)/3+2pin