What is the general solution for cos^4(x)=cos^4(x)csc(x) ?

The general solution for cos^4(x)=cos^4(x)csc(x) is x= pi/2+2pin,x=(3pi)/2+2pin

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cos^4(x)=cos^4(x)csc(x)

Solution

cos^{4} (x) = cos^{4} (x) csc (x)

Solution

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

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

cos^{4} (x) = cos^{4} (x) csc (x)

Subtract

cos^{4} (x) csc (x)

from both sides

cos^{4} (x) - cos^{4} (x) csc (x) = 0

Factor

cos^{4} (x) - cos^{4} (x) csc (x) : - cos^{4} (x) (csc (x) - 1)

cos^{4} (x) - cos^{4} (x) csc (x)

Factor out common term

- cos^{4} (x)

= - cos^{4} (x) (- 1 + csc (x))

- cos^{4} (x) (csc (x) - 1) = 0

Solving each part separately

cos^{4} (x) = 0 or csc (x) - 1 = 0

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

cos^{4} (x) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

cos (x) = 0

The solution is

cos (x) = 0

General solutions for

cos (x) = 0

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

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

csc (x) - 1 = 0 : x = \frac{π}{2} + 2 πn

csc (x) - 1 = 0

Move

1

to the right side

csc (x) - 1 = 0

Add

1

to both sides

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

Simplify

csc (x) = 1

csc (x) = 1

General solutions for

csc (x) = 1

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos^4(x)=cos^4(x)csc(x) ?
The general solution for cos^4(x)=cos^4(x)csc(x) is x= pi/2+2pin,x=(3pi)/2+2pin