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

The general solution for (cot(a))/(cos(a))=cos(a) is No Solution for a\in\mathbb{R}

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(cot(a))/(cos(a))=cos(a)

Solution

\frac{cot ( a )}{cos ( a )} = cos (a)

Solution

No Solution for a \in R

Solution steps

\frac{cot ( a )}{cos ( a )} = cos (a)

Subtract

cos (a)

from both sides

\frac{cot ( a )}{cos ( a )} - cos (a) = 0

Simplify

\frac{cot ( a )}{cos ( a )} - cos (a) : \frac{cot ( a ) - cos ^{2} ( a )}{cos ( a )}

\frac{cot ( a )}{cos ( a )} - cos (a)

Convert element to fraction:

cos (a) = \frac{cos ( a ) cos ( a )}{cos ( a )}

= \frac{cot ( a )}{cos ( a )} - \frac{cos ( a ) cos ( a )}{cos ( a )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{cot ( a ) - cos ( a ) cos ( a )}{cos ( a )}

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

cot (a) - cos (a) cos (a)

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

cos (a) cos (a)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= cos^{1 + 1} (a)

Add the numbers:

1 + 1 = 2

= cos^{2} (a)

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

= \frac{cot ( a ) - cos ^{2} ( a )}{cos ( a )}

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Express with sin, cos

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

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= - cos^{2} (a) + \frac{cos ( a )}{sin ( a )}

Simplify

- cos^{2} (a) + \frac{cos ( a )}{sin ( a )} : \frac{- cos ^{2} ( a ) sin ( a ) + cos ( a )}{sin ( a )}

- cos^{2} (a) + \frac{cos ( a )}{sin ( a )}

Convert element to fraction:

cos^{2} (a) = \frac{cos ^{2} ( a ) sin ( a )}{sin ( a )}

= - \frac{cos ^{2} ( a ) sin ( a )}{sin ( a )} + \frac{cos ( a )}{sin ( a )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- cos ^{2} ( a ) sin ( a ) + cos ( a )}{sin ( a )}

= \frac{- cos ^{2} ( a ) sin ( a ) + cos ( a )}{sin ( a )}

\frac{cos ( a ) - cos ^{2} ( a ) sin ( a )}{sin ( a )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos (a) - cos^{2} (a) sin (a) = 0

Factor

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

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

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

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

= cos (a) - cos (a) cos (a)

Factor out common term

cos (a)

= cos (a) (1 - sin (a) cos (a))

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

Solving each part separately

cos (a) = 0 or 1 - sin (a) cos (a) = 0

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

cos (a) = 0

General solutions for

cos (a) = 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}

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

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

1 - sin (a) cos (a) = 0 :

No Solution

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

Rewrite using trig identities

1 - sin (a) cos (a)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= 1 - \frac{sin ( 2 a )}{2}

1 - \frac{sin ( 2 a )}{2} = 0

Move

1

to the right side

1 - \frac{sin ( 2 a )}{2} = 0

Subtract

1

from both sides

1 - \frac{sin ( 2 a )}{2} - 1 = 0 - 1

Simplify

- \frac{sin ( 2 a )}{2} = - 1

- \frac{sin ( 2 a )}{2} = - 1

Multiply both sides by

2

- \frac{sin ( 2 a )}{2} = - 1

Multiply both sides by

2

2 (- \frac{sin ( 2 a )}{2}) = 2 (- 1)

Simplify

- sin (2 a) = - 2

- sin (2 a) = - 2

Divide both sides by

- 1

- sin (2 a) = - 2

Divide both sides by

- 1

\frac{- sin ( 2 a )}{- 1} = \frac{- 2}{- 1}

Simplify

sin (2 a) = 2

sin (2 a) = 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

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

Since the equation is undefined for:

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

No Solution for a \in R

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Frequently Asked Questions (FAQ)

What is the general solution for (cot(a))/(cos(a))=cos(a) ?
The general solution for (cot(a))/(cos(a))=cos(a) is No Solution for a\in\mathbb{R}