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

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

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Popular >

cot^2(a)=cos^2(a)(cot(a)cos(a))^2

Solution

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

Solution

a = \frac{π}{2} + πn

Degrees

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

Solution steps

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

Subtract

cos^{2} (a) (cot (a) cos (a))^{2}

from both sides

cot^{2} (a) - cos^{4} (a) cot^{2} (a) = 0

Factor

cot^{2} (a) - cos^{4} (a) cot^{2} (a) : - cot^{2} (a) (cos^{2} (a) + 1) (cos (a) + 1) (cos (a) - 1)

cot^{2} (a) - cos^{4} (a) cot^{2} (a)

Factor out common term

- cot^{2} (a)

= - cot^{2} (a) (- 1 + cos^{4} (a))

Factor

cos^{4} (a) - 1 : (cos^{2} (a) + 1) (cos (a) + 1) (cos (a) - 1)

cos^{4} (a) - 1

Rewrite

cos^{4} (a) - 1

(cos^{2} (a))^{2} - 1^{2}

cos^{4} (a) - 1

Rewrite

1

1^{2}

= cos^{4} (a) - 1^{2}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

cos^{4} (a) = (cos^{2} (a))^{2}

= (cos^{2} (a))^{2} - 1^{2}

= (cos^{2} (a))^{2} - 1^{2}

Apply Difference of Two Squares Formula:

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

(cos^{2} (a))^{2} - 1^{2} = (cos^{2} (a) + 1) (cos^{2} (a) - 1)

= (cos^{2} (a) + 1) (cos^{2} (a) - 1)

Factor

cos^{2} (a) - 1 : (cos (a) + 1) (cos (a) - 1)

cos^{2} (a) - 1

Rewrite

1

1^{2}

= cos^{2} (a) - 1^{2}

Apply Difference of Two Squares Formula:

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

cos^{2} (a) - 1^{2} = (cos (a) + 1) (cos (a) - 1)

= (cos (a) + 1) (cos (a) - 1)

= (cos^{2} (a) + 1) (cos (a) + 1) (cos (a) - 1)

= - cot^{2} (a) (cos^{2} (a) + 1) (cos (a) + 1) (cos (a) - 1)

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

Solving each part separately

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

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

cot^{2} (a) = 0

Apply rule

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

cot (a) = 0

General solutions for

cot (a) = 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

a = \frac{π}{2} + πn

a = \frac{π}{2} + πn

cos^{2} (a) + 1 = 0 :

No Solution

cos^{2} (a) + 1 = 0

Solve by substitution

cos^{2} (a) + 1 = 0

Let:

cos (a) = u

u^{2} + 1 = 0

u^{2} + 1 = 0 : u = i, u = - i

u^{2} + 1 = 0

Move

1

to the right side

u^{2} + 1 = 0

Subtract

1

from both sides

u^{2} + 1 - 1 = 0 - 1

Simplify

u^{2} = - 1

u^{2} = - 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - 1, u = - - 1

Simplify

- 1 : i

- 1

Apply imaginary number rule:

- 1 = i

= i

Simplify

- - 1 : - i

- - 1

Apply imaginary number rule:

- 1 = i

= - i

u = i, u = - i

Substitute back

u = cos (a)

cos (a) = i, cos (a) = - i

cos (a) = i, cos (a) = - i

cos (a) = i :

No Solution

cos (a) = i

No Solution

cos (a) = - i :

No Solution

cos (a) = - i

No Solution

Combine all the solutions

No Solution

cos (a) + 1 = 0 : a = π + 2 πn

cos (a) + 1 = 0

Move

1

to the right side

cos (a) + 1 = 0

Subtract

1

from both sides

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

Simplify

cos (a) = - 1

cos (a) = - 1

General solutions for

cos (a) = - 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}

a = π + 2 πn

a = π + 2 πn

cos (a) - 1 = 0 : a = 2 πn

cos (a) - 1 = 0

Move

1

to the right side

cos (a) - 1 = 0

Add

1

to both sides

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

Simplify

cos (a) = 1

cos (a) = 1

General solutions for

cos (a) = 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}

a = 0 + 2 πn

a = 0 + 2 πn

Solve

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn

Combine all the solutions

a = \frac{π}{2} + πn, a = π + 2 πn, a = 2 πn

Since the equation is undefined for:

π + 2 πn, 2 πn

a = \frac{π}{2} + πn

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4 sin^{2} (x) = 8 sin^{2} (\frac{x}{2})

(cos(x)-3sin(x))/(sin(x)+3cos(x))=-1/2

\frac{cos ( x ) - 3 sin ( x )}{sin ( x ) + 3 cos ( x )} = - \frac{1}{2}

cos(pi/2-x)=0

cos (\frac{π}{2} - x) = 0

1sin(45)=2.42sin(r)

1 sin (4 5^{\circ}) = 2.42 sin (r)

Frequently Asked Questions (FAQ)

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