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

The general solution for (sec(x))/(tan(x))=(cos(x))/((sin^2(x))) is x= pi/4+pin

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(sec(x))/(tan(x))=(cos(x))/((sin^2(x)))

Solution

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

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

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

from both sides

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

Simplify

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

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

Least Common Multiplier of

tan (x), sin^{2} (x) : sin^{2} (x) tan (x)

tan (x), sin^{2} (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

tan (x)

sin^{2} (x)

= sin^{2} (x) tan (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

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

For

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

multiply the denominator and numerator by

sin^{2} (x)

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

For

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

multiply the denominator and numerator by

tan (x)

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x)

= - sin (x) + sin^{2} (x) sec (x)

- sin (x) + sec (x) sin^{2} (x) = 0

Factor

- sin (x) + sec (x) sin^{2} (x) : sin (x) (- 1 + sec (x) sin (x))

- sin (x) + sec (x) sin^{2} (x)

Apply exponent rule:

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

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

= - sin (x) + sin (x) sin (x)

Factor out common term

sin (x)

= sin (x) (- 1 + sec (x) sin (x))

sin (x) (- 1 + sec (x) sin (x)) = 0

Solving each part separately

sin (x) = 0 or - 1 + sec (x) sin (x) = 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

- 1 + sec (x) sin (x) = 0 : x = \frac{π}{4} + πn

- 1 + sec (x) sin (x) = 0

Rewrite using trig identities

- 1 + sec (x) sin (x)

sec (x) sin (x) = tan (x)

sec (x) sin (x)

Express with sin, cos

sec (x) sin (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{cos ( x )} sin (x)

Simplify

\frac{1}{cos ( x )} sin (x) : \frac{sin ( x )}{cos ( x )}

\frac{1}{cos ( x )} sin (x)

Multiply fractions:

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

= \frac{1 sin ( x )}{cos ( x )}

Multiply:

1 \cdot sin (x) = sin (x)

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

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

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

Use the basic trigonometric identity:

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

= tan (x)

= - 1 + tan (x)

- 1 + tan (x) = 0

Move

1

to the right side

- 1 + tan (x) = 0

Add

1

to both sides

- 1 + tan (x) + 1 = 0 + 1

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

Since the equation is undefined for:

2 πn, π + 2 πn

x = \frac{π}{4} + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for (sec(x))/(tan(x))=(cos(x))/((sin^2(x))) ?
The general solution for (sec(x))/(tan(x))=(cos(x))/((sin^2(x))) is x= pi/4+pin