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Popular Trigonometry >

cot^2(x)sec(x)-sec(x)=2cot^2(x)-2

Solution

cot^{2} (x) sec (x) - sec (x) = 2 cot^{2} (x) - 2

Solution

x = \frac{3 π}{4} + πn, x = \frac{π}{4} + πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

cot^{2} (x) sec (x) - sec (x) = 2 cot^{2} (x) - 2

Subtract

2 cot^{2} (x) - 2

from both sides

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

Factor

cot^{2} (x) sec (x) - sec (x) - 2 cot^{2} (x) + 2 : (cot (x) + 1) (cot (x) - 1) (sec (x) - 2)

cot^{2} (x) sec (x) - sec (x) - 2 cot^{2} (x) + 2

= (cot^{2} (x) sec (x) - sec (x)) + (- 2 cot^{2} (x) + 2)

Factor out

- 2

from

- 2 cot^{2} (x) + 2 : - 2 (cot^{2} (x) - 1)

- 2 cot^{2} (x) + 2

Factor out common term

- 2

= - 2 (cot^{2} (x) - 1)

Factor out

sec (x)

from

cot^{2} (x) sec (x) - sec (x) : sec (x) (cot^{2} (x) - 1)

cot^{2} (x) sec (x) - sec (x)

Factor out common term

sec (x)

= sec (x) (cot^{2} (x) - 1)

= - 2 (cot^{2} (x) - 1) + sec (x) (cot^{2} (x) - 1)

Factor out common term

cot^{2} (x) - 1

= (cot^{2} (x) - 1) (sec (x) - 2)

Factor

cot^{2} (x) - 1 : (cot (x) + 1) (cot (x) - 1)

cot^{2} (x) - 1

Rewrite

1

1^{2}

= cot^{2} (x) - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

cot^{2} (x) - 1^{2} = (cot (x) + 1) (cot (x) - 1)

= (cot (x) + 1) (cot (x) - 1)

= (cot (x) + 1) (cot (x) - 1) (sec (x) - 2)

(cot (x) + 1) (cot (x) - 1) (sec (x) - 2) = 0

Solving each part separately

cot (x) + 1 = 0 or cot (x) - 1 = 0 or sec (x) - 2 = 0

cot (x) + 1 = 0 : x = \frac{3 π}{4} + πn

cot (x) + 1 = 0

Move

1

to the right side

cot (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

cot (x) = - 1

cot (x) = - 1

General solutions for

cot (x) = - 1

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

cot (x) - 1 = 0 : x = \frac{π}{4} + πn

cot (x) - 1 = 0

Move

1

to the right side

cot (x) - 1 = 0

Add

1

to both sides

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

Simplify

cot (x) = 1

cot (x) = 1

General solutions for

cot (x) = 1

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

sec (x) - 2 = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sec (x) - 2 = 0

Move

2

to the right side

sec (x) - 2 = 0

Add

2

to 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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{3 π}{4} + πn, x = \frac{π}{4} + πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Graph

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