What is the general solution for tan(x)cos(x)+tan(x)=0 ?

The general solution for tan(x)cos(x)+tan(x)=0 is x=pin

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tan(x)cos(x)+tan(x)=0

tan (x) cos (x) + tan (x) = 0

x = πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n

Solution steps

tan (x) cos (x) + tan (x) = 0

Factor

tan (x) cos (x) + tan (x) : tan (x) (cos (x) + 1)

tan (x) cos (x) + tan (x)

Factor out common term

tan (x)

= tan (x) (cos (x) + 1)

tan (x) (cos (x) + 1) = 0

Solving each part separately

tan (x) = 0 or cos (x) + 1 = 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

cos (x) + 1 = 0 : x = π + 2 πn

cos (x) + 1 = 0

Move

1

to the right side

cos (x) + 1 = 0

Subtract

1

from both sides

cos (x) + 1 - 1 = 0 - 1

Simplify

cos (x) = - 1

cos (x) = - 1

General solutions for

cos (x) = - 1

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

Combine all the solutions

x = πn, x = π + 2 πn

Merge Overlapping Intervals

x = πn

sin (θ) = 508

2 = csc (x) cot (x)

sin^{2} (x) = 5

sin^{2} (x) = a

θ, sin (2 θ) = cos (θ)

What is the general solution for tan(x)cos(x)+tan(x)=0 ?
The general solution for tan(x)cos(x)+tan(x)=0 is x=pin