What is the general solution for 1/(tan(x))+(cos^2(x))/(sin(x))=0 ?

The general solution for 1/(tan(x))+(cos^2(x))/(sin(x))=0 is No Solution for x\in\mathbb{R}

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1/(tan(x))+(cos^2(x))/(sin(x))=0

Solution

\frac{1}{tan ( x )} + \frac{cos ^{2} ( x )}{sin ( x )} = 0

Solution

No Solution for x \in R

Solution steps

\frac{1}{tan ( x )} + \frac{cos ^{2} ( x )}{sin ( x )} = 0

Simplify

\frac{1}{tan ( x )} + \frac{cos ^{2} ( x )}{sin ( x )} : \frac{sin ( x ) + cos ^{2} ( x ) tan ( x )}{tan ( x ) sin ( x )}

\frac{1}{tan ( x )} + \frac{cos ^{2} ( x )}{sin ( x )}

Least Common Multiplier of

tan (x), sin (x) : tan (x) sin (x)

tan (x), sin (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

tan (x)

sin (x)

= tan (x) sin (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

tan (x) sin (x)

For

\frac{1}{tan ( x )} :

multiply the denominator and numerator by

sin (x)

\frac{1}{tan ( x )} = \frac{1 \cdot sin ( x )}{tan ( x ) sin ( x )} = \frac{sin ( x )}{tan ( x ) sin ( x )}

For

\frac{cos ^{2} ( x )}{sin ( x )} :

multiply the denominator and numerator by

tan (x)

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

= \frac{sin ( x )}{tan ( x ) sin ( x )} + \frac{cos ^{2} ( x ) tan ( x )}{sin ( x ) tan ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( x ) + cos ^{2} ( x ) tan ( x )}{tan ( x ) sin ( x )}

\frac{sin ( x ) + cos ^{2} ( x ) tan ( x )}{tan ( x ) sin ( x )} = 0

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

sin (x) + cos^{2} (x) tan (x) = 0

Rewrite using trig identities

sin (x) + cos^{2} (x) tan (x)

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x) cos (x)

= sin (x) + sin (x) cos (x)

sin (x) + cos (x) sin (x) = 0

Factor

sin (x) + cos (x) sin (x) : sin (x) (cos (x) + 1)

sin (x) + cos (x) sin (x)

Factor out common term

sin (x)

= sin (x) (1 + cos (x))

sin (x) (cos (x) + 1) = 0

Solving each part separately

sin (x) = 0 or cos (x) + 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

cos (x) + 1 = 0 : x = π + 2 πn

cos (x) + 1 = 0

Move

1

to the right side

cos (x) + 1 = 0

Subtract

1

from 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 = π + 2 πn

x = π + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for 1/(tan(x))+(cos^2(x))/(sin(x))=0 ?
The general solution for 1/(tan(x))+(cos^2(x))/(sin(x))=0 is No Solution for x\in\mathbb{R}