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

The general solution for tan(x)sec(x)=tan(x) is x=pin

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tan(x)sec(x)=tan(x)

tan (x) sec (x) = tan (x)

x = πn

Degrees

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

Solution steps

tan (x) sec (x) = tan (x)

Subtract

tan (x)

from both sides

tan (x) sec (x) - tan (x) = 0

Factor

tan (x) sec (x) - tan (x) : tan (x) (sec (x) - 1)

tan (x) sec (x) - tan (x)

Factor out common term

tan (x)

= tan (x) (sec (x) - 1)

tan (x) (sec (x) - 1) = 0

Solving each part separately

tan (x) = 0 or sec (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

sec (x) - 1 = 0 : x = 2 πn

sec (x) - 1 = 0

Move

1

to the right side

sec (x) - 1 = 0

Add

1

to both sides

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

Simplify

sec (x) = 1

sec (x) = 1

General solutions for

sec (x) = 1

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 = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Combine all the solutions

x = πn, x = 2 πn

Merge Overlapping Intervals

x = πn

1 - sin (2 x) = cos (2 x)

sin (2 x - 5 \frac{π}{6}) = - \frac{1}{2}, 0 \leq x \leq 2 π

sin (x) = 1, 0 \leq x \leq 2 π

6 csc (θ) - 6 = 0

9 sin (x) tan (x) + 4 tan (x) = 0

What is the general solution for tan(x)sec(x)=tan(x) ?
The general solution for tan(x)sec(x)=tan(x) is x=pin