What is the general solution for cot(θ)+tan(θ)=0 ?

The general solution for cot(θ)+tan(θ)=0 is No Solution for θ\in\mathbb{R}

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cot(θ)+tan(θ)=0

Solution

cot (θ) + tan (θ) = 0

Solution

No Solution for θ \in R

Solution steps

cot (θ) + tan (θ) = 0

Rewrite using trig identities

cot (θ) + tan (θ)

Use the basic trigonometric identity:

tan (x) = \frac{1}{cot ( x )}

= cot (θ) + \frac{1}{cot ( θ )}

cot (θ) + \frac{1}{cot ( θ )} = 0

Solve by substitution

cot (θ) + \frac{1}{cot ( θ )} = 0

Let:

cot (θ) = u

u + \frac{1}{u} = 0

u + \frac{1}{u} = 0 : u = i, u = - i

u + \frac{1}{u} = 0

Multiply both sides by

u

u + \frac{1}{u} = 0

Multiply both sides by

u

uu + \frac{1}{u} u = 0 \cdot u

Simplify

uu + \frac{1}{u} u = 0 \cdot u

Simplify

uu : u^{2}

uu

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

\frac{1}{u} u : 1

\frac{1}{u} u

Multiply fractions:

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

= \frac{1 \cdot u}{u}

Cancel the common factor:

u

= 1

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u^{2} + 1 = 0

u^{2} + 1 = 0

u^{2} + 1 = 0

Solve

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

u = i, u = - i

Substitute back

u = cot (θ)

cot (θ) = i, cot (θ) = - i

cot (θ) = i, cot (θ) = - i

cot (θ) = i :

No Solution

cot (θ) = i

No Solution

cot (θ) = - i :

No Solution

cot (θ) = - i

No Solution

Combine all the solutions

No Solution for θ \in R

Popular Examples

cot(θ)+tan(θ)=1

cot (θ) + tan (θ) = 1

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sec (θ) = \frac{1}{2}

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sin(1/5 x)=0

sin (\frac{1}{5} x) = 0

sin(x)= 3/(2sqrt(3))

sin (x) = \frac{3}{2 3}

Frequently Asked Questions (FAQ)

What is the general solution for cot(θ)+tan(θ)=0 ?
The general solution for cot(θ)+tan(θ)=0 is No Solution for θ\in\mathbb{R}