What is the general solution for cot^2(x)cos(x)=cot^2(x) ?

The general solution for cot^2(x)cos(x)=cot^2(x) is x= pi/2+pin

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cot^2(x)cos(x)=cot^2(x)

cot^{2} (x) cos (x) = cot^{2} (x)

x = \frac{π}{2} + πn

Degrees

x = 9 0^{\circ} + 18 0^{\circ} n

Solution steps

cot^{2} (x) cos (x) = cot^{2} (x)

Subtract

cot^{2} (x)

from both sides

cot^{2} (x) cos (x) - cot^{2} (x) = 0

Factor

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

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

Factor out common term

cot^{2} (x)

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

cot^{2} (x) (cos (x) - 1) = 0

Solving each part separately

cot^{2} (x) = 0 or cos (x) - 1 = 0

cot^{2} (x) = 0 : x = \frac{π}{2} + πn

cot^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

cot (x) = 0

General solutions for

cot (x) = 0

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}{2} + πn

x = \frac{π}{2} + πn

cos (x) - 1 = 0 : x = 2 πn

cos (x) - 1 = 0

Move

1

to the right side

cos (x) - 1 = 0

Add

1

to 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 = 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 = \frac{π}{2} + πn, x = 2 πn

Since the equation is undefined for:

2 πn

x = \frac{π}{2} + πn

4 sin^{2} (θ) - 8 cos (θ) + 1 = 0, 0 \leq θ < 2 π

4 sin (A) + 3 = 2 sin (A) + 4

0 = - 18 cos (3 x)

cos^{2} (2 x) - 3 cos (2 x) = 0

csc^{2} (x) = sin (x)

What is the general solution for cot^2(x)cos(x)=cot^2(x) ?
The general solution for cot^2(x)cos(x)=cot^2(x) is x= pi/2+pin